Let $~y_{1}(t)$,$~y_{2}(t)$ be solutions of the diff. eq. $y ''+ e^{t}y '+y = t^{3} $ Which of the following is a solution of the ODE? Let $~y_{1}(t)$,$~y_{2}(t)$ be solutions of the diff. eq. $y ''+ e^{t}y '+y = t^{3} $ Which of the following is a solution of the ODE?


*

*$y_{1}(t)-y_{2}(t)$ 

*$y_{1}(t)+y_{2}(t)$

*$y_{1}(t)-5y_{2}(t)$ 

*$2y_{1}(t)-y_{2}(t)$ 

*$2y_{1}(t)+3y_{2}(t)$ 
I'm not sure how to approach this problem and was hoping someone would be willing to assist.

We have $y_i''+e^ty_i'+y_i=t^3$ for $i=1,2$. Suppose $y(t)=my_1(t)+ny_2(t)$ is a solution of the ODE, $m,n$ constant. Plug this into the ODE to get:$$m(y_1''+e^ty_1'+y_1)+n(y_2''+e^ty_2'+y_2)=t^3$$This gives $(m+n)t^3=t^3$, that is $m+n=1$. The correct answer is...?
