# Solving non-homogeneous linear second-order differential equation with repeated roots

I have the following second-order linear differential equation that I am unable to solve: $y''+4y'+4y=t$. The method for solving homogeneous linear second-order differential equations obviously doesn't work, but I am unsure how to apply the technique to solve non-homogenous linear second-order differential equations because this equation has repeated roots. Is there a particular method I should be applying?

## 2 Answers

A particular solution is obviously a degree $1$ polynomial: $y_0=at+b$. Since $y_0'=a$ and $y_0''=0$, the relation yields $4a+at+b=t$, that is, $a=1$ and $b=-4$.

For the general solution the general method does work! When the characteristic polynomial has a root $\lambda$ of multiplicity $m$, you get linearly independent solutions of the form $t^{k}e^{\lambda t}$, for $k=0,1,\dots,m-1$.

In this case, you get $y_1=e^{-2t}$ and $y_2=te^{-2t}$.

one solution is given by $$C_1e^{-2t}$$ and you must for the second one $$C_2te^{-2t}$$ since both solutions must be independend