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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.

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  • $\begingroup$ 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? $\endgroup$
    – Nasser
    Commented Oct 17, 2021 at 20:40
  • $\begingroup$ @Nasser I was never introduced to any vector/matrix based way of solving DEs; this is just a differential equations question. $\endgroup$
    – user
    Commented Oct 18, 2021 at 2:05
  • $\begingroup$ 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. $\endgroup$
    – Nasser
    Commented Oct 18, 2021 at 2:16

1 Answer 1

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Verify that

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

so you now can substitute and solve separately.

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