0

Find the general solution of the differential equations

$$\left \{\begin{matrix}\dfrac{dx}{dt} & = & x-2y, \\ \dfrac{dy}{dt} & = & 2x+5y.\end{matrix}\right .$$

So far, I was able to find the eigenvalues to be $3$ (with multiplicity $2$). Also, I think the eigenvector is $(-1,1)$.

commie trivial
  • 984
  • 2
  • 4
  • 14
chickenpotpie2004
  • 51
  • 1
    $$x' + y' = (x + y)' = 3(x + y) \implies \dots$$ If you want to do it via linear algebra methods, you'll need to find a second eigenvector of the system. – Matthew Cassell Apr 18 '23 at 03:12
  • @WillJagy I didn't say independent. – Matthew Cassell Apr 18 '23 at 04:06
  • $$ \left( \begin{array}{rr} -2 & 0 \ 2 & 1 \
    \end{array} \right) \left( \begin{array}{rr} -\frac{1}{2} & 0 \ 1 & 1 \
    \end{array} \right) = \left( \begin{array}{rr} 1 & 0 \ 0 & 1 \
    \end{array} \right) $$

    $$ \left( \begin{array}{rr} -2 & 0 \ 2 & 1 \
    \end{array} \right) \left( \begin{array}{rr} 3 & 1 \ 0 & 3 \
    \end{array} \right) \left( \begin{array}{rr} -\frac{1}{2} & 0 \ 1 & 1 \
    \end{array} \right) = \left( \begin{array}{rr} 1 & -2 \ 2 & 5 \
    \end{array} \right) $$

     – Will Jagy Apr 18 '23 at 16:03

0 Answers0