Differential Equations/Transform Theory
 Variable separable form
Basic form : f (x) dx + g(y) dy = 0
Solution :
 Linear differential equations
Basic form : = Q
where P and Q are function of x only
Solution :
is called I.F
âˆ´ Solution : y (IF) =
 Homogeneous differential equation
Basic form :
Take y = vx
Solution :
To find solution for the equation.
 Auxiliary equation, AE = D^{n} + P, D^{nâ€“}^{1}+ ...... + P_{n} = 0
Solve for D and find roots m_{1}, m_{2},......
 Obtain the complementary function as follows:
Roots of AE

CF


Real and different roots (m_{1}, m_{2}, ... all unequal)



Real roots and two of them equal (m_{1}= m_{2}, m_{3}, m_{4},...)



A pair of imaginary roots (Î± + iÎ², Î± â€“ iÎ², m_{3}, m_{4} ...)


(c_{1} cos Î²x + c_{2} sin Î²x) e^{Î±x} +


pairs of equal imaginary roots (Î± Â± iÎ², Î± Â± iÎ², m_{5}, m_{6} ...)


TIPS :
If f (a) = 0
PI =
If f â€²(a) = 0
PI =
 Find the particular integral (PI)
PI =
Case 1:
When X = e^{ax}
PI = ; f (a) â‰ 0
Case 2:
When X = sin (ax + b)
PI = Put D^{2} = â€“ a^{2}
Case 3:
When X = cos (ax + b)
PI = Put b^{2} = â€“ a^{2}
Case 4:
When X = X^{m}
PI =
Find [f (D)]^{â€“1} by binomial theorem in ascending powers of D.
Case 5:
When x = e^{ax} . V (V is a function f_{x})
PI =
=
Then find value of as in above cases.
Case 6:
When X is a functions of x
PI =
Resolve into partial fractions and operate each partial fraction on X by using
 The general solution is :
y = CF + PI
Cauchyâ€™s homogeneous linear equation
Equation is : + k_{n}y = X
X is a function of x.
This can be reduced to linear DE with constant coeff. by substituting x = e^{t} or log x = t
Then = Dy = D(D â€“ 1) y etc.