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I missed a week of class because I was sick and I'm trying to do my homework but I simply don't know how. I tried looking for videos on youtube but couldn't find anything similar (not sure what these are called? Differential system of equations maybe?). Any explanation on the steps needed to solve these problems (or the one below) would be greatly appreciated. Please note I'm looking for how to solve them, not the answer (answers are already in the back of the book). Thanks!

Solve the systems \begin{align*} \frac{\mathrm{d}x}{\mathrm{d}t} &= 2x -y\\ \frac{\mathrm{d}y}{\mathrm{d}t} &= x \end{align*}

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    $\begingroup$ Are you familiar with finding eigenvalues of a matrix? That is do you have a basis in linear algebra? $\endgroup$
    – Triatticus
    Apr 18 '17 at 17:00
  • $\begingroup$ Google for "solving a system of first order linear differential equations". For example, this might help: tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx. $\endgroup$
    – levap
    Apr 18 '17 at 17:00
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    $\begingroup$ Does your class use a textbook? $\endgroup$
    – MPW
    Apr 18 '17 at 17:01
  • $\begingroup$ Thanks for the help guys. The class is called differential equations with linear algebra, so it's a little bit of both (we must have just started the linear algebra part). There isn't a textbook unfortunately. I am going over Paul's notes right now. $\endgroup$ Apr 18 '17 at 17:17
  • $\begingroup$ Write it in matrix form, left multiply both sides by a matrix $M$ with row vectors $(a,b)$ and $(c,d)$ so that $MA=M$, where $A$ is the given matrix. Then make the substitution $z=ax+by$ and $w=cx+dy$ and you get $z=e^t+c_1$, $w=e^t+c_2$. $\endgroup$
    – Aravind
    Apr 18 '17 at 17:23
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Hints:

Method 1: Take the derivative of the first equation, substitute in the second equation, solve. You have

$$x'' = 2 x' - y' = 2 x' - x$$

Method 2: Find the eigenvalues / eigenvectors. You will arrive at (note that the second eigenvector is a generalized one):

$$\lambda_{1, 2} = 1, v_1 = (1,1), v_2 = (1,0)$$

the solution will be written as:

$$X(t) = c_1 e^{\lambda_1 t} (t v_1 + v_2) + c_2 e^{\lambda_2 t} v_1= c_1 e^t\left( t\begin{pmatrix} 1 \\ 1 \\ \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \\ \end{pmatrix}\right) + c_2 e^t \begin{pmatrix} 1 \\ 1 \\ \end{pmatrix}$$

Method 3: Laplace transforms

Method n: Many other methods, like Matrix Exponential, Fundamental Matrix...

You can search out many examples of each on MSE.

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  • $\begingroup$ Variant of Method 2: Don’t bother with finding generalized eigenvectors and use the exponential of $A-I$ directly. $\endgroup$
    – amd
    Apr 18 '17 at 18:27
  • $\begingroup$ @amd: Good point! $\endgroup$
    – Moo
    Apr 18 '17 at 18:28
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HINT

By second equation, we have: $$y = \int x(t)dt + C$$

Put it back to first equation you get: $$x' = 2x- \int x(t)dt - C$$ $$x'' = 2x' - x$$

All derivatives are with respect to $t$

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  • $\begingroup$ The R.H.S. shouldn't be $2-x$ because the derivatives are w.r..t $t$, so instead of 2, you get $2(2x-y)$, which doesn't help. $\endgroup$
    – Aravind
    Apr 18 '17 at 17:08
  • $\begingroup$ @Aravind very good point! I've edited my answer. But I do not see where you get $2(2x-y)$ though.. $\endgroup$
    – Jay Zha
    Apr 18 '17 at 17:10

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