# How to solve this system of differential equations? $\dot{x}=3x-3y+5e^{2t},\ \dot{y}=3x-3y$

I am trying to solve the following system of 2nd order differential equations ($$\dot x$$ denotes the derivative with respect to $$t$$, $$x$$ and $$y$$ are functions of $$t$$): $$\begin{cases} \dot{x}=3x-3y+5e^{2t} &(1)\\ \dot{y}=3x-3y &(2) \end{cases}$$

So far, I have done the following:

Differentiating both sides of $$(1)$$ with respect to $$t$$, $$\ddot{x}=3\dot x -3\dot y +10e^{2t}$$ Substituting $$\dot y$$, $$\ddot{x}=3\dot x -9x+9y+10e^{2t}$$ $$\ddot{x}-3\dot x +9x = 9y+10e^{2t}$$ Doing the same for $$y$$ gives $$\ddot y+3\dot y+9y = 9x+15e^{2t}$$

I don't know where to go further with this. I don't think I can use the method of undetermined coefficients here since the RHS is a function of both $$t$$ and another variable ($$x$$ or $$y$$).

Wolfram Alpha gives a neat (not messy) solution to the problem, so I assume we don't need some crazy methods to solve this.

• I think you are supposed to use the eigenvalue/eigenvector method to solve such problems as a systematic way which always works. Sometimes one can find short cuts. But if this was in a text book, was it in the linear systems section of the textbook where you are expected to use eigenvectors? Commented Oct 17, 2021 at 20:40
• @Nasser I was never introduced to any vector/matrix based way of solving DEs; this is just a differential equations question.
– user
Commented Oct 18, 2021 at 2:05
• Well, this is the standard method actually to solve a system of first order ode's. Here is example solving-system-of-linear-differential-equations-by-eigenvalues and many more on the net. This is a very important method as it works on any system of linear odes'. You just need to find the eigenvalues and eigenvector of the A matrix and you get the solution. Commented Oct 18, 2021 at 2:16

$$\dot x-\dot y = 5e^{2t}\Rightarrow x-y = \frac 52 e^{2t}+ C_0$$