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Differential Equations/Transform Theory

  1. Variable separable form
     
    Basic form : f (xdx g(y) dy = 0
     
    Solution : Description: 435.png
     
  2. Linear differential equations
     
    Basic form : Description: 440.png = Q
     
    where P and Q are function of x only
     
    Solution : Description: 445.png
     
    Description: 450.png is called I.F
     
     Solution : y (IF) = Description: 455.png
     
  3.  Homogeneous differential equation
     
    Basic form : Description: 460.png
     
    Take y = vx
     
    Solution : Description: 465.png 
To find solution for the equation.
  1. Auxiliary equation, AE = Dn + P, Dn–1+ ...... + Pn = 0
     
    Solve for D and find roots m1m2,......
     
  2. Obtain the complementary function as follows:

Roots of AE

CF

  1. Real and different roots (m1m2, ... all unequal)
  • Description: 470.png
  1. Real roots and two of them equal (m1m2m3m4,...)
  • Description: 475.png
  1. A pair of imaginary roots (α + iβα – iβ, m3m4 ...)
  • (c1 cos βx + c2 sin βxeαx + Description: 480.png
  1. pairs of equal imaginary roots (α ± iβα ± iβ, m5m6 ...)
  • Description: 485.png
 
TIPS :
If f (a) = 0
PI = Description: 490.png 
If f ′(a) = 0
PI = Description: 495.png 
 
  1. Find the particular integral (PI)
     
    PI = Description: 500.png
     
    Case 1: 
     
    When X = eax
     
    PI = Description: 505.pngf (a) ≠ 0
     
    Case 2: 
     
    When X = sin (ax + b)
     
    PI = Description: 510.png Put D2 = – a2
     
    Case 3: 
     
    When X = cos (ax + b)
     
    PI = Description: 515.png Put b2 = – a2
     
    Case 4: 
     
    When X = Xm
     
    PI = Description: 520.png 
     
    Find [f (D)]–1 by binomial theorem in ascending powers of D.
     
    Case 5: 
     
    When x = eax . V (V is a function fx)
     
    PI = Description: 525.png 
     
    = Description: 530.png 
     
    Then find value of Description: 536.png as in above cases.
     
    Case 6: 
     
    When X is a functions of x
     
    PI = Description: 541.png 
     
    Resolve Description: 546.png into partial fractions and operate each partial fraction on X by using
     
    Description: 551.png  
  2. The general solution is :
     
    y = CF + PI 
Cauchy’s homogeneous linear equation
Equation is : Description: 556.png kny = X
X is a function of x.
This can be reduced to linear DE with constant coeff. by substituting x = et or log x = t
Then Description: 561.png = Dy Description: 566.png = D(D – 1) y etc.





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