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Mathematics
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Session
Differential Equations - 1
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Session Objectives
Differential Equation
Order and Degree
Solution of a Differential Equation, Generaland Particular Solution
Initial Value Problems
Formation of Differential Equations
Class Exercise
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Differential Equation
An equation containing an independentvariable x,dependent variable y and thedifferential coefficients of the dependentvariable y with respect to independentvariable x, i.e.
2
2
dy d y, ,
dx dx
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Examples
dy1 = 3xydx
2
2
d y2 + 4y = 0
dx
3
3 2
3 2
d y d y dy3 + + + 4y = si x
dxdx dx
2 24 x dx + y dy = 0
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Order of the Differential Equation
The order of a differential equation is the orderof the highest order derivative occurring in thedifferential equation.
2 32 2
3 2 2
d y dy d y dyExam le : = =dx dxdx dx
The order of the highest order derivative
2
2d y is 2 .dx
Therefore, order is 2
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Degree of the Differential Equation
The degree of a differential equation is the degree of the highestorder derivative, when differential coefficients are ade free frofractions and radicals.
3322 22 22
2 2
d y dy d y dyExa le : + 1+ = 0 = 1+
dx dxdx dx
The degree of the highest order derivative is 2.
2
2d ydx
Therefore, degree is 2.
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Example - 1
Deter ine the order and degree of the differential
equation:d d
= +a 1+d d
.
2dy dy
Sol tion: W y = x + a 1+dx dx
2dy dy
y - x = a 1+dx dx
2 22dy dyy - x = a 1+
dx dx
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Solution Cont.
2 22 2 2 2dy dy dyy - 2xy + x = a + a
dx dx dx
The order of the highest derivative isand its degree is 2.
dydx
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Example - 2
Deter ine the order and degree of the differential
equation:
324 2
4
d y dyc
dxdx
!
Solution: e have
3
322 24 42
4 4
d y dy d y dy= c + = c +
dx dxdx dx
Here, the order of the highest order is 4
4
4
d y
dx
and, the degree of the highest order is 24
4
d y
dx
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Linear and Non-LinearDifferential Equation
A differential equation in which the dependent variable y and
its differential coefficients i.e. occur only in the
first degree and are not ultiplied together is called a lineardifferential equation. Otherwise, it is a non-linear differentialequation.
2
2
dy d y, ,
dx dx
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Example - 3
is a linear differential equation of order 2 and degree .
is a non-linear differential equation because the dependent
variable y and its derivative are ultiplied together.dy
dx
2
2
d y dyi - 7y 4x
dxdx
dyii y - 4 xdx
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Solution of a Differential Equation
The solution of a differential equation is the relationbetween the variables, not taking the differentialcoefficients, satisfying the given differential equation andcontaining as any arbitrary constants as its order is.
For exa ple: y Acosx - Bsinx
is a solution of the differential equation
2
2
d y4y 0
dx
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General Solution
If the solution of a differential equation of nth order contains narbitrary constants, the solution is called the general solution.
is the general solution of the differential equation 22
d y y 0dx
y B sin x!
is not the general solution as it contains one arbitrary constant.
y Acosx - Bsinx
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Particular Solution
A solution obtained by giving particular values to the arbitraryconstants in general solution is called particular solution.
y 3 cos 2 si!
is a particular solution of the differential equation
2
2
d yy 0.
dx
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Example - 4
3 2Solution: e have y x ax bx c (i)
2dy 3x 2ax b (ii) Differentiating i w.r.t. xdx
2
2
d y6x 2a (iii) Differentiation ii w.r.t. x
dx
3 2
3
3
Verify t at y x ax bx i a olution of t e
d ydifferential e uation 6.
dx
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Solution Cont.
3
3
d y= 6 Diff r ntiating iii .r.t x
dx
3
3
d y= 6 is a diff r ntial ation of i .
dx
@
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Initial Value Problems
The proble in which we find the solution of thedifferential equation that satisfies so e prescribedinitial conditions, is called initial value proble .
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Example - 5
2x x
2
dy d ye , e
dx dx
xy e@ satisfies the differential equation2
2
d y dy- 0
dxdx
Show that is the solution of the initial value
proble
xy e
2
2
d y dy- 0 y 0 2 y' 0
dxdx
xSolution : We have y e
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Solution Cont.
0 0
x
0
dyy 0 e and e
dx
y 0 2 and y 0 1 !
x
y e@is the solution of the initial value proble .
x dyy e anddx
xe@ !
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Formation of Differential Equations
y x
Assu e the fa ily of straightlines represented by
dy
dx
dy y
dx x
dyx ydx
@ !
!
is a differential equation of the first order.
X
Y
O
U m = tanU
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Formation of Differential Equations
Assu e the fa ily of curves represented by
where A and B are arbitrary constants.
y = A co s x + (i)
dy A si ... iidx
@ ! [Differentiating (i) w.r.t. x]
2
2
d ya n d A co s x B
d x
! [Differentiating (ii) w.r.t. x]
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Formation of Differential Equations
2
2
d yy
dx ! [Using (i)]
2
2
d yy 0
dx
is a differential equation of second order
Si ilarly, by eli inating three arbitrary constants, a differentialequation of third order is obtained.
Hence, by eli inating n arbitrary constants, a differentialequation of nth order is obtained.
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Example - 6
For the differential equation of the fa ily of curves
a and c being para eters. y = a sin bx + c ,
Solution: e have y = a sin bx + c
is the required differential equation.
2 22 2
2 2
d y d y= -b y + b y = 0
dx dx
[Differentiating w.r.t. x] dy = ab cos bx + cdx
[Differentiating w.r.t. x] 2
2
2
d y= -ab sin bx + c
dx
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Example - 7
Find the differential equation of the fa ily of allthe circles, which passes through the origin andwhose centre lies on the y-axis.
If it passes through (0, 0), we get c 0
2 2x y 2gx 2y 0@
This is an equation of a circle with centre (- g, - f)and passing through (0, 0).
Solution: The general equation of a circle is
2 2x y 2gx 2y c 0.
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Solution Cont.
Now if centre lies on y-axis, then g 0.
2 2x + y + 2y 0 (i)@
This represents the required fa ily of circles.
dyx y
dxdy
dx
!
dy dy2x 2y 2 0 Differentiating i w.r.t. xdx dx
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Solution Cont.
? A2 2
dyx y dxx y - 2y 0 Substituting the value of f
dy
dx
2 2 2dy dy
x y - 2xy - 2y 0dx dx
2 2 dyx - y - 2xy 0dx
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Thank you