Use the method of undetermined coefficients to find the gene

Use the method of undetermined coefficients to find the general solution of the Aifteteotial equation y\" - 10y\' + 25y = e^5t + e-^5t.

Solution

First we solve the homogeneous ODE

y\'\'-10y\'+25y=0

Assuming y=e^{kt} and substituting gives

k^2-10k+25=0

k=5 ie repeated roots

So,

y=e^{5t}(A+Bt)

This is the complementary solution.

Now let us find the particular solution.

So since e^{5t} is already solution to homogeneous equation we take guess for particular solution as

C t^2 e^{5t}+D e^{-5t}

Substituting gives

(5Ct^2e^{5t}+2Cte^{5t})

e^{5t}(20Ct+25Ct^2+2C)-10(5Ct^2+2Ct)e^{5t}+25Ct^2 e^{5t}

+25 De^{-5t}+50 De^{-5t}+25 D e^{-5t}=e^{5t}+e^{-5t}

Comparing coefficients gives

2C=1 hence, C=1/2

100 D=1 so D=1/100

This gives us the general solution