ma 108 - ordinary differential equationsdey/diffeqn_spring14/lecture7_d2.pdf · santanu dey lecture...
TRANSCRIPT
![Page 1: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/1.jpg)
MA 108 - Ordinary Differential Equations
Santanu Dey
Department of Mathematics,Indian Institute of Technology Bombay,
Powai, Mumbai [email protected]
March 10, 2014
Santanu Dey Lecture 7
![Page 2: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/2.jpg)
Outline of the lecture
Lipschitz continuity
Existence & uniqueness
Picard’s iteration
Second order linear equations
Santanu Dey Lecture 7
![Page 3: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/3.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.
1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 4: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/4.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) =
1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 5: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/5.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt =
1 +x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 6: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/6.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 7: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/7.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) =
1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 8: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/8.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt
= 1 +x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 9: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/9.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 10: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/10.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) =
1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 11: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/11.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n.
(By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 12: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/12.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) =
limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 13: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/13.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x)
= ex2/2.
Santanu Dey Lecture 7
![Page 14: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/14.jpg)
Example : Picard’s
Solve : y ′ = xy , y(0) = 1 using Picard’s iteration method.1 The integral equation is
y(x) = 1 +
∫ x
x0
ty dt.
2 The successive approximations are :
y1(x) = 1 +
∫ x
0t · 1 dt = 1 +
x2
2.
y2(x) = 1 +
∫ x
0t(1 +
t2
2) dt = 1 +
x2
2+
x4
2 · 4.
...
yn(x) = 1 + (x2
2) +
1
2!(x2
2)2 + · · ·+ 1
n!(x2
2)n. (By induction )
3 y(x) = limn→∞
yn(x) = ex2/2.
Santanu Dey Lecture 7
![Page 15: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/15.jpg)
Exercises
1 Does uniform continuity =⇒ Lipschitz continuity ?
(No, consider f (x) =√x x ∈ [0, 2].)
2 The value of n such that the curves xn + yn = C are theorthogonal trajectories of the family
y =x
1− Kx
is . . . . . . . . .? (n=3).
Santanu Dey Lecture 7
![Page 16: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/16.jpg)
Exercises
1 Does uniform continuity =⇒ Lipschitz continuity ?(No, consider f (x) =
√x x ∈ [0, 2].)
2 The value of n such that the curves xn + yn = C are theorthogonal trajectories of the family
y =x
1− Kx
is . . . . . . . . .? (n=3).
Santanu Dey Lecture 7
![Page 17: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/17.jpg)
Exercises
1 Does uniform continuity =⇒ Lipschitz continuity ?(No, consider f (x) =
√x x ∈ [0, 2].)
2 The value of n such that the curves xn + yn = C are theorthogonal trajectories of the family
y =x
1− Kx
is . . . . . . . . .?
(n=3).
Santanu Dey Lecture 7
![Page 18: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/18.jpg)
Exercises
1 Does uniform continuity =⇒ Lipschitz continuity ?(No, consider f (x) =
√x x ∈ [0, 2].)
2 The value of n such that the curves xn + yn = C are theorthogonal trajectories of the family
y =x
1− Kx
is . . . . . . . . .? (n=3).
Santanu Dey Lecture 7
![Page 19: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/19.jpg)
Exercises
1 Does uniform continuity =⇒ Lipschitz continuity ?(No, consider f (x) =
√x x ∈ [0, 2].)
2 The value of n such that the curves xn + yn = C are theorthogonal trajectories of the family
y =x
1− Kx
is . . . . . . . . .? (n=3).
Santanu Dey Lecture 7
![Page 20: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/20.jpg)
Uniqueness of solution
Suppose φ1 and φ2 are both solutions of
y ′ = f (x , y), y(x0) = y0 .
Thus, both these satisfy the integral equation
φi (x) = y0 +
∫ x
x0
f (t, φi (t)) dt i = 1, 2.
For x ≥ x0,
φ1(x)− φ2(x) =
∫ x
x0
(f (t, φ1(t))− f (t, φ2(t))) dt.
Thus,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t))− f (t, φ2(t))| dt.
Since f satisfies Lipschitz condition w.r.t. the second variable, wehave
|f (t, φ1(t))− f (t, φ2(t))| ≤ M|φ1(t)− φ2(t)|.
Santanu Dey Lecture 7
![Page 21: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/21.jpg)
Uniqueness of solution
Suppose φ1 and φ2 are both solutions of
y ′ = f (x , y), y(x0) = y0 .
Thus, both these satisfy the integral equation
φi (x) = y0 +
∫ x
x0
f (t, φi (t)) dt i = 1, 2.
For x ≥ x0,
φ1(x)− φ2(x) =
∫ x
x0
(f (t, φ1(t))− f (t, φ2(t))) dt.
Thus,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t))− f (t, φ2(t))| dt.
Since f satisfies Lipschitz condition w.r.t. the second variable, wehave
|f (t, φ1(t))− f (t, φ2(t))| ≤ M|φ1(t)− φ2(t)|.
Santanu Dey Lecture 7
![Page 22: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/22.jpg)
Uniqueness of solution
Suppose φ1 and φ2 are both solutions of
y ′ = f (x , y), y(x0) = y0 .
Thus, both these satisfy the integral equation
φi (x) = y0 +
∫ x
x0
f (t, φi (t)) dt i = 1, 2.
For x ≥ x0,
φ1(x)− φ2(x) =
∫ x
x0
(f (t, φ1(t))− f (t, φ2(t))) dt.
Thus,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t))− f (t, φ2(t))| dt.
Since f satisfies Lipschitz condition w.r.t. the second variable, wehave
|f (t, φ1(t))− f (t, φ2(t))| ≤ M|φ1(t)− φ2(t)|.
Santanu Dey Lecture 7
![Page 23: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/23.jpg)
Uniqueness of solution
Suppose φ1 and φ2 are both solutions of
y ′ = f (x , y), y(x0) = y0 .
Thus, both these satisfy the integral equation
φi (x) = y0 +
∫ x
x0
f (t, φi (t)) dt i = 1, 2.
For x ≥ x0,
φ1(x)− φ2(x) =
∫ x
x0
(f (t, φ1(t))− f (t, φ2(t))) dt.
Thus,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t))− f (t, φ2(t))| dt.
Since f satisfies Lipschitz condition w.r.t. the second variable, wehave
|f (t, φ1(t))− f (t, φ2(t))| ≤ M|φ1(t)− φ2(t)|.
Santanu Dey Lecture 7
![Page 24: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/24.jpg)
Uniqueness of solution
Suppose φ1 and φ2 are both solutions of
y ′ = f (x , y), y(x0) = y0 .
Thus, both these satisfy the integral equation
φi (x) = y0 +
∫ x
x0
f (t, φi (t)) dt i = 1, 2.
For x ≥ x0,
φ1(x)− φ2(x) =
∫ x
x0
(f (t, φ1(t))− f (t, φ2(t))) dt.
Thus,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t))− f (t, φ2(t))| dt.
Since f satisfies Lipschitz condition w.r.t. the second variable, wehave
|f (t, φ1(t))− f (t, φ2(t))| ≤ M|φ1(t)− φ2(t)|.
Santanu Dey Lecture 7
![Page 25: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/25.jpg)
Uniqueness of solution
Suppose φ1 and φ2 are both solutions of
y ′ = f (x , y), y(x0) = y0 .
Thus, both these satisfy the integral equation
φi (x) = y0 +
∫ x
x0
f (t, φi (t)) dt i = 1, 2.
For x ≥ x0,
φ1(x)− φ2(x) =
∫ x
x0
(f (t, φ1(t))− f (t, φ2(t))) dt.
Thus,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t))− f (t, φ2(t))| dt.
Since f satisfies Lipschitz condition w.r.t. the second variable, wehave
|f (t, φ1(t))− f (t, φ2(t))| ≤ M|φ1(t)− φ2(t)|.
Santanu Dey Lecture 7
![Page 26: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/26.jpg)
Uniqueness of solution
Suppose φ1 and φ2 are both solutions of
y ′ = f (x , y), y(x0) = y0 .
Thus, both these satisfy the integral equation
φi (x) = y0 +
∫ x
x0
f (t, φi (t)) dt i = 1, 2.
For x ≥ x0,
φ1(x)− φ2(x) =
∫ x
x0
(f (t, φ1(t))− f (t, φ2(t))) dt.
Thus,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t))− f (t, φ2(t))| dt.
Since f satisfies Lipschitz condition w.r.t. the second variable, wehave
|f (t, φ1(t))− f (t, φ2(t))| ≤ M|φ1(t)− φ2(t)|.
Santanu Dey Lecture 7
![Page 27: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/27.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 28: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/28.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt
(1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 29: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/29.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt.
Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 30: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/30.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0,
U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 31: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/31.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 32: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/32.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 33: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/33.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 34: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/34.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 35: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/35.jpg)
Uniqueness - proof contd..
That is,
|φ1(x)− φ2(x)| ≤∫ x
x0
|f (t, φ1(t)− f (t, φ2(t)| dt
≤∫ x
x0
M|φ1(t)− φ2(t)| dt (1)
Set U(x) =
∫ x
x0
|φ1(t)− φ2(t)| dt. Then,
U(x0) = 0, U(x) ≥ 0, ∀x ≥ x0.
Further, U(x) is differentiable and
U ′(x) = |φ1(x)− φ2(x)|.
Hence, (1) yieldsU ′(x)−MU(x) ≤ 0.
Santanu Dey Lecture 7
![Page 36: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/36.jpg)
Uniqueness Proof contd...
Multiplying the above equation by e−Mx gives
(e−MxU(x))′ ≤ 0 for x ≥ x0.
Integrating this from x0 to x we get U(x) ≤ 0 for x ≥ x0. So
U(x) = 0 ∀x ≥ x0 =⇒ U ′(x) ≡ 0 =⇒ φ1(x) ≡ φ2(x)
which contradicts the initial hypothesis.Use a similar argument to show for x ≤ x0.
Thus, φ1(x) ≡ φ2(x).
Santanu Dey Lecture 7
![Page 37: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/37.jpg)
Uniqueness Proof contd...
Multiplying the above equation by e−Mx gives
(e−MxU(x))′ ≤ 0 for x ≥ x0.
Integrating this from x0 to x we get U(x) ≤ 0 for x ≥ x0. So
U(x) = 0 ∀x ≥ x0
=⇒ U ′(x) ≡ 0 =⇒ φ1(x) ≡ φ2(x)
which contradicts the initial hypothesis.Use a similar argument to show for x ≤ x0.
Thus, φ1(x) ≡ φ2(x).
Santanu Dey Lecture 7
![Page 38: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/38.jpg)
Uniqueness Proof contd...
Multiplying the above equation by e−Mx gives
(e−MxU(x))′ ≤ 0 for x ≥ x0.
Integrating this from x0 to x we get U(x) ≤ 0 for x ≥ x0. So
U(x) = 0 ∀x ≥ x0 =⇒ U ′(x) ≡ 0
=⇒ φ1(x) ≡ φ2(x)
which contradicts the initial hypothesis.Use a similar argument to show for x ≤ x0.
Thus, φ1(x) ≡ φ2(x).
Santanu Dey Lecture 7
![Page 39: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/39.jpg)
Uniqueness Proof contd...
Multiplying the above equation by e−Mx gives
(e−MxU(x))′ ≤ 0 for x ≥ x0.
Integrating this from x0 to x we get U(x) ≤ 0 for x ≥ x0. So
U(x) = 0 ∀x ≥ x0 =⇒ U ′(x) ≡ 0 =⇒ φ1(x) ≡ φ2(x)
which contradicts the initial hypothesis.Use a similar argument to show for x ≤ x0.
Thus, φ1(x) ≡ φ2(x).
Santanu Dey Lecture 7
![Page 40: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/40.jpg)
Uniqueness Proof contd...
Multiplying the above equation by e−Mx gives
(e−MxU(x))′ ≤ 0 for x ≥ x0.
Integrating this from x0 to x we get U(x) ≤ 0 for x ≥ x0. So
U(x) = 0 ∀x ≥ x0 =⇒ U ′(x) ≡ 0 =⇒ φ1(x) ≡ φ2(x)
which contradicts the initial hypothesis.Use a similar argument to show for x ≤ x0.
Thus, φ1(x) ≡ φ2(x).
Santanu Dey Lecture 7
![Page 41: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/41.jpg)
Summary - First Order Equations
Linear Equations
- Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 42: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/42.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 43: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/43.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 44: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/44.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 45: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/45.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separable
Reducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 46: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/46.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separable
Exact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 47: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/47.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factors
Reducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 48: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/48.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 49: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/49.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 50: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/50.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theorem
Picard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 51: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/51.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 52: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/52.jpg)
Summary - First Order Equations
Linear Equations - Solution
Reducible to linear - Bernoulli
Non-linear equations
Variable separableReducible to variable separableExact equations - Integrating factorsReducible to Exact
Existence & Uniqueness results for IVP :
y ′ = f (x , y), y(x0) = y0
Peano’s existence theoremPicard’s existence-uniqueness theorem
Picard’s iteration method
Santanu Dey Lecture 7
![Page 53: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/53.jpg)
Second order differential equations
Recall that a general second order linear ODE is of the form
a2(x)d2y
dx2+ a1(x)
dy
dx+ a0(x)y = g(x).
An ODE of the form
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x)
is called a second order linear ODE in standard form.Though there is no formula to find all the solutions of such anODE, we study the existence, uniqueness and number of solutionsof such ODE’s.
Santanu Dey Lecture 7
![Page 54: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/54.jpg)
Second order differential equations
Recall that a general second order linear ODE is of the form
a2(x)d2y
dx2+ a1(x)
dy
dx+ a0(x)y = g(x).
An ODE of the form
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x)
is called a second order linear ODE in standard form.
Though there is no formula to find all the solutions of such anODE, we study the existence, uniqueness and number of solutionsof such ODE’s.
Santanu Dey Lecture 7
![Page 55: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/55.jpg)
Second order differential equations
Recall that a general second order linear ODE is of the form
a2(x)d2y
dx2+ a1(x)
dy
dx+ a0(x)y = g(x).
An ODE of the form
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x)
is called a second order linear ODE in standard form.Though there is no formula to find all the solutions of such anODE, we study the existence, uniqueness and number of solutionsof such ODE’s.
Santanu Dey Lecture 7
![Page 56: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/56.jpg)
Homogeneous Linear Second Order DE
If r(x) ≡ 0 in the equation
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x),
that is,d2y
dx2+ p(x)
dy
dx+ q(x)y = 0,
then the ODE is said to be homogeneous.Otherwise it is called non-homogeneous.
Santanu Dey Lecture 7
![Page 57: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/57.jpg)
Homogeneous Linear Second Order DE
If r(x) ≡ 0 in the equation
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x),
that is,d2y
dx2+ p(x)
dy
dx+ q(x)y = 0,
then the ODE is said to be homogeneous.Otherwise it is called non-homogeneous.
Santanu Dey Lecture 7
![Page 58: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/58.jpg)
Homogeneous Linear Second Order DE
If r(x) ≡ 0 in the equation
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x),
that is,d2y
dx2+ p(x)
dy
dx+ q(x)y = 0,
then the ODE is said to be homogeneous.
Otherwise it is called non-homogeneous.
Santanu Dey Lecture 7
![Page 59: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/59.jpg)
Homogeneous Linear Second Order DE
If r(x) ≡ 0 in the equation
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x),
that is,d2y
dx2+ p(x)
dy
dx+ q(x)y = 0,
then the ODE is said to be homogeneous.Otherwise it is called non-homogeneous.
Santanu Dey Lecture 7
![Page 60: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/60.jpg)
Homogeneous Linear Second Order DE
If r(x) ≡ 0 in the equation
d2y
dx2+ p(x)
dy
dx+ q(x)y = r(x),
that is,d2y
dx2+ p(x)
dy
dx+ q(x)y = 0,
then the ODE is said to be homogeneous.Otherwise it is called non-homogeneous.
Santanu Dey Lecture 7
![Page 61: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/61.jpg)
Initial Value Problem- Existence/Uniqueness
An initial value problem of a second order homogeneous linearODE is of the form:
y ′′ + p(x)y ′ + q(x)y = 0, y(x0) = a, y ′(x0) = b,
where p(x) and q(x) are assumed to be continuous on an openinterval I with x0 ∈ I , has a unique solution y(x) in the interval I .
Santanu Dey Lecture 7
![Page 62: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/62.jpg)
Initial Value Problem- Existence/Uniqueness
An initial value problem of a second order homogeneous linearODE is of the form:
y ′′ + p(x)y ′ + q(x)y = 0,
y(x0) = a, y ′(x0) = b,
where p(x) and q(x) are assumed to be continuous on an openinterval I with x0 ∈ I , has a unique solution y(x) in the interval I .
Santanu Dey Lecture 7
![Page 63: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/63.jpg)
Initial Value Problem- Existence/Uniqueness
An initial value problem of a second order homogeneous linearODE is of the form:
y ′′ + p(x)y ′ + q(x)y = 0, y(x0) = a,
y ′(x0) = b,
where p(x) and q(x) are assumed to be continuous on an openinterval I with x0 ∈ I , has a unique solution y(x) in the interval I .
Santanu Dey Lecture 7
![Page 64: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/64.jpg)
Initial Value Problem- Existence/Uniqueness
An initial value problem of a second order homogeneous linearODE is of the form:
y ′′ + p(x)y ′ + q(x)y = 0, y(x0) = a, y ′(x0) = b,
where p(x) and q(x) are assumed to be continuous on an openinterval I with x0 ∈ I , has a unique solution y(x) in the interval I .
Santanu Dey Lecture 7
![Page 65: MA 108 - Ordinary Differential Equationsdey/diffeqn_spring14/lecture7_D2.pdf · Santanu Dey Lecture 7. Outline of the lecture Lipschitz continuity Existence & uniqueness Picard’s](https://reader035.vdocuments.site/reader035/viewer/2022070108/602579f32db7493b2e2218fc/html5/thumbnails/65.jpg)
Initial Value Problem- Existence/Uniqueness
An initial value problem of a second order homogeneous linearODE is of the form:
y ′′ + p(x)y ′ + q(x)y = 0, y(x0) = a, y ′(x0) = b,
where p(x) and q(x) are assumed to be continuous on an openinterval I with x0 ∈ I , has a unique solution y(x) in the interval I .
Santanu Dey Lecture 7