Use the method of undetermined coefficients to find the solu

Use the method of undetermined coefficients to find the solution of the initial value problem

y\'\'+6y\'=3e^-6t, y(0)=12, y\'(0) = 0

Solution

y\'\'+6y\'=3e^-6t

is the ode to be solved

In method of undetermined coefficients we first solve homogeneous equation

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

Assume, y=exp(kt)

Substituting gives

k^2+6k=0

k=0,-6

So general solution to homogeneous equation is

y=Ae^{0*t}+Be^{-6t}=A+Be^{-6t}

Now we need to look for particular solution. In method of undetermined coefficients we make a guess for particular solution based on the inhomogeneous part which is here

3e^-6t

which is solution to homogeneous equation. So the guess would be

yp=Cte^{-6t}

Substituting gives

6C e^(-6 t) (1-6 t)+12 Ce^(-6 t) (-1+3 t)=3e^{-6t}

6C e^{-6t}-12 Ce^{-6t}=3e^{-6t}

This gives

C=-1/2

y=A+Be^{-6t}-te^{-6t}/2