Download - Constant-Coefficient Systems(1)
โ Then the corresponding solutions are
โ The general solution is
โ Wronskian of (2) is
Constant-Coefficient Systems(2)
โ We can graph solutions of (1), ๐ฒ ๐ก =๐ฆ1(๐ก)๐ฆ2(๐ก)
as a single curve in the
๐ฆ1- ๐ฆ2 plane.
โ That curve is a parametric representation with parameter ๐ก, and
called a trajectory of (1).
โ The ๐ฆ1- ๐ฆ2 plane is called the phase plane.
โ If we fill the phase plane with trajectories, we obtain the so-called
phase portrait of (1)
How to graph solutions in the phase plane
๐ฒโฒ = ๐๐ฒโ๐ฆ1โฒ = ๐11๐ฆ1 + ๐12๐ฆ2
๐ฆ2โฒ = ๐21๐ฆ1 + ๐22๐ฆ2
โฏ(1)
โ By substituting ๐ฒ = ๐ฑ๐๐๐ก and ๐ฒโฒ = ๐๐ฑ๐๐๐ก, we get ๐๐ฑ = ๐๐ฑ.
โ Characteristic equation
โ det(๐ โ ๐๐) =โ3 โ ๐ 1
1 โ3 โ ๐= ๐2 + 6๐ + 8 = 0
โ ๐1 = โ2, ๐2 = โ4
โ For ๐1 = โ2, ๐ฑ(๐) = ๐ ๐ ๐ป, and for ๐2 = โ4, ๐ฑ(๐) = ๐ โ๐ ๐ป
โ The general solution is
Phase portrait : example (1)
๐ฒโฒ = ๐๐ฒ =โ3 11 โ3
๐ฒ โ๐ฆ1โฒ = โ3๐ฆ1 + ๐ฆ2๐ฆ2โฒ = ๐ฆ1 โ 3๐ฆ2
๐ฒ =๐ฆ1๐ฆ2
= ๐1๐ฒ(1) + ๐2๐ฒ
(2) = ๐111๐โ2๐ก + ๐2
1โ1
๐โ4๐ก
โฏ(1)
โ Phase portrait of (1)
โ The point ๐ฒ = ๐ is a common point of all
projectories.
โ Unique tangent direction for every point (๐ฆ1, ๐ฆ2)
except for the point (0, 0).
โ This point (0, 0), at which ๐ ๐๐/๐ ๐๐ is undetermined, is called
a critical point.
Phase portrait : example (1)
๐๐ฆ2๐๐ฆ1
=๐ฆ2โฒ๐๐ก
๐ฆ1โฒ๐๐ก
=๐21๐ฆ1 + ๐22๐ฆ2๐11๐ฆ1 + ๐12๐ฆ2
โ Improper node (two distinct real eigenvalues of the same sign)
: a critical point ๐0 at which all the trajectories, except for two of them,
have the same limiting direction of the tangent. The two exceptional
trajectories also have a limiting direction of the tangent at ๐0 which,
however, is different
Five types of critical points
๐ฒโฒ =โ3 11 โ3
๐ฒ
โ ๐ฒ = ๐111๐โ2๐ก + ๐2
1โ1
๐โ4๐ก
โ Proper node (two identical eigenvalues and two linearly independent
eigenvectors)
: a critical point ๐0 at which every trajectory has a definite limiting
direction and for any given direction ๐ at ๐0 there is a trajectory
having ๐ as its limiting direction.
Five types of critical points
๐ฒโฒ =1 00 1
๐ฒ
โ ๐ฒ = ๐110๐๐ก + ๐2
01๐๐ก or ๐1๐ฆ2 = ๐2๐ฆ1
โ Saddle point (two real eigenvalues of opposite signs)
: a critical point ๐0 at which there are two incoming trajectories, two
outgoing trajectories, and all the other trajectories in a
neighborhood of ๐0 bypass ๐0.
Five types of critical points
๐ฒโฒ =1 00 โ1
๐ฒ
โ ๐ฒ = ๐110๐๐ก + ๐2
01๐โ๐ก or ๐ฆ1๐ฆ2 = ๐๐๐๐ ๐ก
โ Center (two purely imaginary conjugate eigenvalues)
: a critical point ๐0 that is enclosed by infinitely many closed
trajectories.
Five types of critical points
๐ฒโฒ =0 1
โ4 0๐ฒ
โ ๐ฒ = ๐112๐
๐2๐๐ก + ๐20โ2๐
๐โ2๐๐ก
or 2๐ฆ12 +
1
2๐ฆ22 = ๐๐๐๐ ๐ก
โ Spiral point (two complex conjugate eigenvalues)
: a critical point ๐0 about which the trajectories spiral, approaching ๐0as ๐ก โ โ (or tracing these spirals in the opposite sense, away from
๐0).
Five types of critical points
๐ฒโฒ =โ1 1โ1 โ1
๐ฒ
โ ๐ฒ = ๐11๐๐(โ1+๐)๐ก + ๐2
1โ๐
๐(โ1โ๐)๐ก
or ๐ = ๐๐โ๐ก where ๐2 = ๐ฆ12 + ๐ฆ2
2
โ Two identical eigenvalues and one eigenvector
โ ๐ฒ(1) =1
โ1๐3๐ก, and let ๐ฒ(2) = ๐ฑ๐ก๐3๐ก + ๐ฎ๐3๐ก
โ Then ๐ฒ 2 โฒ= ๐ฑ๐3๐ก + 3๐ฑ๐ก๐3๐ก + 3๐ฎ๐3๐ก - (*)
๐๐ฒ 2 = ๐๐ฑ๐ก๐3๐ก + ๐๐ฎ๐3๐ก - (**)
โ ๐ โ ๐๐ ๐ฎ = ๐ฑ, ๐ฎ = 0 1 ๐ (linearly independent of ๐ฑ)
โ ๐ฒ = ๐11
โ1๐3๐ก + ๐2
1โ1
๐ก +01
๐3๐ก
Degenerate node
๐ฒโฒ =4 1
โ1 2๐ฒ โ ๐1 = ๐๐ = ๐ = 3, ๐ฑ1 = ๐ฑ2 = ๐ฑ = 1 โ 1 ๐
(*)=(**) where ๐ =4 1
โ1 2
Stability
โ A critical point ๐0 is called stable if all trajectories that at some
instant are close to ๐0 remain close to ๐0 at all future times
โ For every disk ๐ท of radius ํ > 0 with center ๐0 there is a disk ๐ท๐ฟ of
radius ๐ฟ > 0 with center ๐0 such that every trajectory that has a point
๐1(corresponding to ๐ก = ๐ก1 say) in ๐ท has all its points corresponding
to ๐ก โฅ ๐ก1 in ๐ท .
Criteria for critical points & stability
Stable critical point ๐0(The trajectory initiating at ๐1stays in the disk of radius ํ.)
Stable and attractive
critical point ๐0
๐ฒโฒ = ๐๐ฒ =๐11 ๐12๐21 ๐22
๐ฒCh. eqn. det ๐ โ ๐๐ = ๐2 โ ๐11 + ๐22 ๐ + det๐
โ ๐ = ๐11 + ๐22, ๐ = det๐, ฮ = ๐2 โ 4๐
๐ = ๐1 + ๐2, ๐ = ๐1๐2, ฮ = ๐1 โ ๐22
Stability criteria for critical points
Eigenvalue criteria for critical points
Stability chart
Criteria for critical points & stability
Stability of critical points : Example
โ Free motions of a mass on a spring
๐๐ฆโฒโฒ + ๐๐ฆโฒ + ๐๐ฆ = 0
๐ฒโฒ =0 1
โ๐/๐ โ๐/๐๐ฒ
๐ฆ1 = ๐ฆ๐ฆ2 = ๐ฆโฒ
det ๐ โ ๐๐ = ๐2 + ๐/๐ ๐ + ๐/๐ = 0๐ = โ๐/๐, ๐ = ๐/๐, ฮ = ๐/๐ 2 โ 4๐/๐
No damping (๐ = 0)
๐ = 0, ๐ > 0 โ a center
Underdamping (๐2 < 4๐๐)
๐ < 0, ๐ > 0, ฮ < 0โ a stable and attractive spiral point
Critical damping (๐2 = 4๐๐)
๐ < 0, ๐ > 0, ฮ = 0โ a stable and attractive node
Overdamping (๐2 > 4๐๐)
๐ < 0, ๐ > 0, ฮ > 0โ a stable and attractive node
Linearization of nonlinear systems
โ If ๐1 and ๐2 in (1) are continuous and have continuous partial
derivatives in a neighborhood of the critical point ๐0: (0, 0), and if
det ๐ โ 0 in (2), then the kind and stability of the critical point of (1)
are the same as those of the linearized system
โ Exception : ๐ has equal and pure imaginary eigenvalues
โ same kind of critical points of (3) or a spiral point
๐ฒโฒ = ๐(๐ฒ) โ๐ฆ1โฒ = ๐1(๐ฆ1, ๐ฆ2)
๐ฆ2โฒ = ๐2(๐ฆ1, ๐ฆ2)
โฏ (1)Nonlinear
system
๐ฒโฒ = ๐๐ฒ + ๐ก(๐ฒ) โ๐ฆ1โฒ = ๐11๐ฆ1 + ๐12๐ฆ2 + โ1(๐ฆ1, ๐ฆ2)
๐ฆ2โฒ = ๐21๐ฆ1 + ๐22๐ฆ2 + โ2(๐ฆ1, ๐ฆ2)
โฏ(3)๐ฒโฒ = ๐๐ฒ โ๐ฆ1โฒ = ๐11๐ฆ1 + ๐12๐ฆ2
๐ฆ2โฒ = ๐21๐ฆ1 + ๐22๐ฆ2
Linearization : Example
Free undamped pendulum
Step 1 : modeling
Step 2 : critical points ยฑ2๐๐, 0 ๐ โถ ๐๐๐ก๐๐๐๐
โ Consider linearization near 0, 0 , and set ๐ = ๐ฆ1, ๐โฒ = ๐ฆ2
โ Calculate ๐ = 0, ๐ = ๐, โ= โ4๐ โ a center
โ Due to periodicity, critical points ยฑ2๐๐, 0 (๐ โถ ๐๐๐ก๐๐๐๐) are all
centers.
๐๐ฟ๐โฒโฒ + ๐๐๐ ๐๐๐ = 0 โ ๐โฒโฒ + ๐๐ ๐๐๐ = 0 (๐ = ๐/๐ฟ)
๐โฒโฒ + ๐๐ ๐๐๐ = 0โ๐ฆ1โฒ = ๐ฆ2
๐ฆ2โฒ = โ๐๐ฆ1 (๐ ๐๐๐ฆ1 โ ๐ฆ1)
Linearization : Example
Step 3 : critical points ยฑ(2๐ โ 1)๐, 0 ๐ โถ ๐๐๐ก๐๐๐๐
โ Consider linearization near ๐, 0 , and set ๐ โ ๐ = ๐ฆ1, (๐ โ ๐)โฒ= ๐ฆ2
โ Calculate ๐ = 0, ๐ = โ๐, โ= 4๐ โ a saddle point
โ Due to periodicity, critical points ยฑ(2๐ โ 1)๐, 0 (๐ โถ ๐๐๐ก๐๐๐๐) are
all saddle points.
๐โฒโฒ + ๐๐ ๐๐๐ = 0โ๐ฆ1โฒ = ๐ฆ2
๐ฆ2โฒ = ๐๐ฆ1 (๐ ๐๐๐ฆ1 โ โ๐ฆ1)
Transformation to a 1st order eq. in the phase plane
โ 2nd order autonomous ODE(an ODE in which ๐ก does not occur explicitly)
โ Take ๐ฆ = ๐ฆ1 as the independent variable, set ๐ฆโฒ = ๐ฆ2 and transform ๐ฆโฒโฒ
by the chain rule.
โ Then, the ODE becomes of first order.
๐น ๐ฆ, ๐ฆโฒ, ๐ฆโฒโฒ = 0
๐ฆโฒโฒ = ๐ฆ2โฒ =
๐๐ฆ2๐๐ก
=๐๐ฆ2๐๐ฆ1
๐๐ฆ1๐๐ก
=๐๐ฆ2๐๐ฆ1
๐ฆ2
๐น ๐ฆ1, ๐ฆ2,๐๐ฆ2๐๐ฆ1
๐ฆ2 = 0
Example
Free undamped pendulum
โ set ๐ = ๐ฆ1, ๐โฒ = ๐ฆ2
โ Using separation of variables, we get
๐๐ฟ๐โฒโฒ + ๐๐๐ ๐๐๐ = 0 โ ๐โฒโฒ + ๐๐ ๐๐๐ = 0 (๐ = ๐/๐ฟ)
๐โฒโฒ + ๐๐ ๐๐๐ = 0 โ๐๐ฆ2
๐๐ฆ1๐ฆ2 = โ๐๐ ๐๐๐ฆ1
1
2๐ฆ22 = ๐๐๐๐ ๐ฆ1 + ๐ถ โ
1
2๐ ๐ฟ๐ฆ2
2 โ๐๐ฟ2๐๐๐๐ ๐ฆ1 = ๐๐ฟ2๐ถ
Nonhomogeneous linear systems of ODEs
Example
Method 1 : method of undetermined coefficients
โ ๐ฒ(โ) = ๐111๐โ2๐ก + ๐2
1โ1
๐โ4๐ก
โ Set ๐ฒ(๐) = ๐ฎ๐ก๐โ2๐ก + ๐ฏ๐โ2๐ก (by modification rule) where ๐ฎ = ๐ 1 1 ๐
โ Substitute ๐ฆ(๐) into (1), then we get
๐ฎ โ 2๐ฏ = ๐๐ฏ +โ62
โ๐๐
โ2๐ฃ12๐ฃ2
=โ3๐ฃ1 + ๐ฃ2
๐ฃ1 โ 3๐ฃ2+
โ62
โ ๐ = โ2, ๐ฃ1 = ๐, ๐ฃ2 = ๐ + 4
โ We can choose any real number ๐, like ๐ = 0
๐ฒโฒ = ๐๐ฒ + ๐ =โ3 11 โ3
๐ฒ +โ62
๐โ2๐ก โฏ(1)
๐ฒ = ๐111๐โ2๐ก + ๐2
1โ1
๐โ4๐ก โ 211๐ก๐โ2๐ก +
04๐โ2๐ก
Nonhomogeneous linear systems of ODEs
Example
Method 2 : method of variation of parameters
โ ๐ฒ(โ) = ๐111๐โ2๐ก + ๐2
1โ1
๐โ4๐ก = ๐โ2๐ก ๐โ4๐ก
๐โ2๐ก โ๐โ4๐ก๐1๐2
= ๐ ๐ก ๐
โ Set ๐ฒ(๐) = ๐ ๐ก ๐ฎ ๐ก and substitute into (1) then we get
๐๐ฎโฒ = ๐ โต ๐โฒ = ๐๐ , ๐ฎโฒ = ๐โ๐๐ =1
2
๐2๐ก ๐2๐ก
๐4๐ก โ๐4๐กโ6๐โ2๐ก
2๐โ2๐ก=
โ2โ4๐2๐ก
๐ฎ = 0๐ก โ2โ4๐2๐ก
โ ๐๐กโ =โ2๐ก
โ2๐2๐ก + 2โด ๐ฒ(๐) = ๐๐ฎ =
โ2๐ก โ 2โ2๐ก + 2
๐โ2๐ก +2โ2
๐โ4๐ก
๐ฒโฒ = ๐๐ฒ + ๐ =โ3 11 โ3
๐ฒ +โ62
๐โ2๐ก โฏ(1)
๐ฒ = ๐111๐โ2๐ก + ๐2
1โ1
๐โ4๐ก โ 211๐ก๐โ2๐ก +
โ22
๐โ2๐ก
โข Homogeneous linear DE w/ variable coeff.
โ Power series method
Frobenius method: extension of power series
โ Power series
--- infinite series (in powers of )
โ If , power series in powers of x
Series Solution of DE
0x x
2
0 0 1 0 2 0
0
m
m
m
a x x a a x x a x x
Center of the series
0 0x
2 3
0 1 2 3
0
m
m
m
a x a a x a x a x
Maclaurin Series
2
0
2 3
0
2 2 4
0
2 1 3 5
0
11
1
1! 2! 3!
1cos 1
2 ! 2! 4!
1sin
2 1 ! 3! 5!
m
m
mx
m
m m
m
m m
m
x x xx
x x xe x
m
x x xx
m
x x xx x
m
โ Represent p(x) and q(x) as a power series of x (or )
โ Assume a sol. in the form of power series
โ Collect like powers of x, and solve for the coefficients.
Power Series Method
0y p x y q x y
0x x
2 3
0 1 2 3
0
m
m
m
y a x a a x a x a x
1 2
1 2 3
0
2 3
m
m
m
y ma x a a x a x
2 2
2 3 4
0
1 2 3 2 4 3
m
m
m
y m m a x a a x a x
โ Insert the power series
โ Collect the like powers of x
โ Solve for each coefficient
โ General sol.
Example: Power Series Method
0y y
2 2
1 2 3 0 1 22 3 0 a a x a x a a x a x
2
1 0 2 1 3 22 3 0 a a a a x a a x
0 01 21 0 2 3, , ,
2 2! 3 3!
a aa aa a a a
2 3
0 012! 3!
xx xy a x a e
โ Nth partial sum
โ Remainder
โ If at , the sequence s0(x), s1(x), s2(x), โฆ converges to a value
โ In case of convergence
--- all sn(x1) with n > N lie between s (x1) - and s (x1) +
Theory of Power Series Method
2
0 1 0 2 0 0 n
n ns x a a x x a x x a x x
1 2
1 0 2 0
n n
n n nR x a x x a x x
1x x
1 1 1 0
0
lim
m
n mn
m
s x s x a x x Value, sum
1 1 1 n nR x s x s x for all n > N
Convergence Interval (1)
โ Always converges at . If it is the only point of convergence, all
other terms except a0 are zeros.
โ Further values of x for convergence
0x x
Practically no interest
Convergence interval
0 x x R
Radius of convergence1
lim
m
mm
Ra
Convergence Interval (2)
If the limit is infinite, the power series only converges at .
โ If the limit is 0, the power series converges for all x.
โ Example: geometric series
--- geometric series converges and represents when
1
1
lim
m
mm
Ra
a
0x x
R
2
0
11
1
m
m
x x xx
1, 1 ma R
1x 1 1 x
Operations on Power Series (1)
โ Term-wise differentiation
โ Term-wise addition
0
0
m
m
m
y x a x x
1
0
0
m
m
m
y x ma x x
2
0
0
1
m
m
m
y x m m a x x
0 x x R
0 0
0 0
,
m m
m m
m m
a x x b x x
0
0
m
m m
m
a b x x
Operations on Power Series (2)
โ Term-wise multiplication
โ Vanishing of all coefficients
--- a positive radius of convergence, a sum is identically zero.
โ Shifting summation indices
0 0
0 0
,
m m
m m
m m
a x x b x x
0 1 1 0 0
0
m
m m m
m
a b a b a b x x
2
0 0 0 1 1 0 0 0 2 1 1 2 0 0 a b a b a b x x a b a b a b x x
2 2 1
2 1
1 2
m m
m m
m m
x m m a x ma x
Operations on Power Series (3)
โ Shifting summation indices (contโd)
โ Set
โ Existence of power series sol.
--- if p(x), q(x), r(x) can be represented as a power series at ,
then every sol. is analytic.
1
2 1
1 2
m m
m m
m m
m m a x ma x
1, 1 s m m s
1
2 0
1 2 1
s s
s s
s s
s s a x s a x
1
2
1 2 1
s
s s
s
s s a s a x
0x x
Legendreโs Eqn. & Polynomials (1)
--- boundary value problems for spheres
โ Sol.: Legendre function
โ Since the coeff.โs are analytic, power series method can be applied.
Apply
21 2 1 0 x y xy n n y
Theory of special functions
0
m
m
m
y a x
2 2 1
0 0 0
1 1 2 0
m m m
m m m
m m m
x m m a x x ma x k a x
Legendreโs Eqn. & Polynomials (2)
โ Arranging each power
โ Coeff. of each power must be zero.
2
0 0
1 1 0
m m
m m
m m
m m a x m m a x
2
2 3 4 22 1 3 2 4 3 2 1 s
sa a x a x s s a x2
22 1 a x
2
1 22 1 2 2 1 s
sa x a x s s a x2
0 1 2 2 0 s
ska ka x ka x sa x
22 1 0 a n n
3 16 2 1 0 a n n a
0x
1x
Legendreโs Eqn. & Polynomials (3)
โ Recurrence relation (formula)
โ a0, a1: arbitrary constants
22 1 1 2 1 0 s ss s a s s s n n a 2,3,s
2
1
2 1
s s
n s n sa a
s s
2 0
3 1
1,
2!
1 2,
3!
n na a
n na a
0,1,s
Legendreโs Eqn. & Polynomials (4)
โ General sol.
โ Converges for
โ y1, y2: linearly independent
0 1 1 2 y a y x a y x
2 4
1
1 2 1 31
2! 4!
n n n n n ny x x
3 5
2
1 2 3 1 2 4
3! 5!
n n n n n ny x x x
1x
Legendre Polynomials (1)
โ n is a nonnegative integer in many applications. Then, when s = n,
an+2 = 0, an+4 = 0, an+6 = 0, ยท ยท ยท.
โ If n is even, y1 reduces to a polynomial of degree n
โ If n is odd, y2 reduces to a polynomial of degree n
โ All the nonvanishing coeff. can be represented in terms of an
โ an is arbitrary, and choose an = 1 when n = 1
Legendre polynomials
2
2 1
1s s
s sa a
n s n s
2s n
2
2 ! 1 3 5 2 1,
!2 !n n
n na
nn
1,2,n
Legendre Polynomials (2)
โ Then,
โ In general, when n โ 2m โฅ 0
โ Pn(x): Legendre polynomial of degree n
2
2 2 !,
2 1 ! 2 !n n
na
n n
4
2 4 !
2 2! 2 ! 4 !n n
na
n n
2
2 2 !1
2 ! ! 2 !
m
n m n
n ma
m n m n m
2
0
2
2
2 2 !1
2 ! ! 2 !
2 ! 2 2 !
2 1! 1 ! 2 !2 !
Mm n m
n nm
n n
nn
n mP x x
m n m n m
n nx x
n nn
Legendre Polynomials (3)
0
2
2
4 2
4
1
3
3
5 3
5
1,
13 1 ,
2
135 30 3 ,
8
1,
15 3 ,
2
163 70 15 ,
8
P x
P x x
P x x x
P x
P x x x
P x x x x
โ First few of the functions