Particular Integral

  1. [1/f(D)]eax = [1/f(a)]eax
    If f(a) = 0 then [1/f(D)]eax = x[1/f'(a)]eax
    If f'(a) = 0 then [1/f(D)]eax = x2[1/f''(a)]eax

  2. [1/f(D)]xn = [f(D)]-1xn expand [f(D)]-1 and then operate

  3. [1/f(D2)]sin ax = [1/f(-a2)]sin ax
    and [1/f(D2)]cos ax = [1/f(-a2)]cos ax
    If f(-a2) = 0 then [1/f(D2)]sin ax = x[1/f'(-a2)]sin ax
    Similar argument for cos.

  4. [1/f(D)]eax φ(x) = eax [1/f(D+a)]φ(x)

  5. [1/(D+a)]φ(x) = e-axeaxφ(x) dx

Particular integral of product of polynomial and trigonometric functions

  • [1/f(D)]xn sin ax
    [1/f(D)]xn(cos ax + i sin ax) = [1/f(D)]xn eiax = eiax[1/f(D+ia)]xn
    [1/f(D)]xn sin ax = Imaginary part of eiax [1/f(D+ia)]xn
    [1/f(D)]xn cos ax = Real part of eiax [1/f(D+ia)]xn

General Method of finding particular integral

General method of finding the particular integral of any function φ(x)

P.I. = [1/(D - a)]φ(x) = y

or

(D -  a)/(D - a) φ(x) = (D - a) y
φ(x) = (D - a) y or φ(x) = Dy - ay

dy/dx - ay =  φ(x) which is a linear differential equation.
Its solution is ye-∫adx = e-∫adxφ(x)dx 
or ye-ax = e-axφ(x)dx
y = eaxe-axφ(x)dx
[1/(D - a)]φ(x) = eaxe-axφ(x)dx


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