Differential Equation Undetermined coeffcient Method d2ydx

Undetermined coeffcient Method d^2y/dx - 4dy/dx + 3y = 3 + e^2x; y(0) = 10, y(0) = -4

Solution

First we solve the homogeneous ODE

y\'\'-4y\'+3y=0

Let, y=e^{kx}

Substituting gives

K^2-4k+3=0

k=1,3

So general solution to inhomogeneous ODE is

yh= a e^x+b e^{3x}

Based on inhomogeneous part the guess for the particular solution

yp= C + D e^{2x}

Substituting gives

4D e^{2x} -8 D e^{2x}+3D e^{2x}+3C=3+ e^{2x}

D=-1, C=1

So,

general solution is

y(x)= a e^x+b e^{3x}+1 - e^{2x}

y(0)=0

Hence, a+b+1-1=0 hence, a=-b

y\'(0)=-1

So,

a+3b-2=1

a+3b=3

b=3/2,a=-3/2

y(x)= -3/2( e^x - e^{3x})+1 - e^{2x}