Chapter 4.7, Problem 40P

In each of problems $40$ and $41$, use the method outlined in problem $39$ to solve the given differential equation:

$t{y}^{″}-\left(1+t\right)y\text{'}+y=t^2{e}^{2t},t>0;y_1\left(t\right)=1+t$.

39. The method of reduction of order (see the discussion preceding problem $28$ in section $4.2$) can also be used for the nonhomogeneous equation

$y\text{'}\text{'}+p\left(t\right)y\text{'}+q\left(t\right)y=g\left(t\right)$,         (i)

Provided one of the solution $y_1$ of the corresponding homogenous equation is known. Let $y=v\left(t\right)y_1\left(t\right)$ and show that $y$ satisfies Eq. (i) if $v$ is a solution of

$y_1\left(t\right)v\text{'}\text{'}+[2y_1\text{'}\left(t\right)+p\left(t\right)y_1\left(t\right)]v\text{'}=g\left(t\right)$.         (ii)

Equation (ii) is a first order linear equation for $v\text{'}$. Solving this equation, integrating the result, and then multiplying by $y_1(t)$ lead to the general solution of Eq. (i).