find a general solution 2 c 8 pts Find a general solution to

find a general solution

2 (c) (8 pts.) Find a general solution to Hint: One solution is yi = e.

Given d^2y/dx^2 - 3. dy/dx - 2y =0

d^2y/dx^2 - 3. dy/dx = 2y

to find a general solution let us try y = aX^3 + bX^2 + CX

dy/dx = 3aX^2 + 2bX + C

d^2y/dx^2 = 6ax+2b

SUBSTITUTING THESE in the given equation

6ax+2b - 3 ( 3aX^2 + 2bX + C) = 2 (aX^3 + bX^2 + CX )

6ax + 2b -9aX^2 - 6bx - 3c = 2aX^3 + 2bx^2 + 2cx

equating two equation we have

9a = 2b ; 6a-6b = 2c ; 2b-3c = 0

from here 3c = 2b where c = 2b/3 ; a = 2b/9

substituting these

6(2b/9) - 6b = 2 ( 2b/3)

4b/3 - 6b = 4b/3

particular integral is given by

YP = (2/9)X^3 + (3/2)X^2 + (2/3)X

YC = C + De^(-x)

General solution Y = YC + YP = (2/9)X^3 + (3/2)X^2 + (2/3)X + C + De^(-x)