y 6y 9y 4e2x y0 4 y0 3SolutionFirst we solve the homoge

y\" + 6y\' + 9y = 4e^2x, y(0) = 4, y\'(0) = 3

Solution

First we solve the homogeneous ode

y\'\'+6y\'+9y=0

Let, y=e^{kx}

Substituting gives

k^2+6k+9=0

THis gives k=-3

So,

y= e^{3x}(A+Bx)

Now for particular solution we make guess based on the inhomogeneous part

yp=C e^{2x}

Substituting gives

4C e^{2x}+12 Ce^{2x}+9C e^{2x}=4 e^{2x}

THis gives, C=4/25

Hence,y= e^{3x}(A+Bx)+4 e^{2x}/25

y(0)=4

So, A+4/25=4

A=96/25

y\'= e^{3x}(B+3A+3Bx)+8 e^{2x}/25

y\'(0)=B+3A+8/25=3=B+288/25+8/25

3=B+296/25

B=-221/25