Question

Решить cистему dx\dt=6*x+3*y dy\dt=-8*x-5*y

Answer 1

Найдем базисные решения в виде

$x = Ae^{\lambda t}\quad y=Be^{\lambda t}\\\\ \left\{ \begin{aligned} &\lambda A = 6A+3B\\ &\lambda B = -8A-5B \end{aligned} \right.\\\\\\ \det \left[\begin{array}{ccc}6-\lambda&3\\-8&-5-\lambda\end{array}\right] =0\\\\ (\lambda+5)(\lambda-6)+24 = 0\\ \lambda^2 -\lambda-6 = 0\\ \lambda_1 = 3\\ \lambda_2 = -2$

Нашли собственные значения, теперь собственные вектора

$1)\\ \left[\begin{array}{cc}3&3\\-8&-8\end{array}\right]\left[\begin{array}{c}A\\B\end{array}\right] =0\\\\ A = -B = C_1 2)\\ \left[\begin{array}{cc}8&3\\-8&-3\end{array}\right]\left[\begin{array}{c}A\\B\end{array}\right] =0\\\\ A = 3C_2\quad B=-8C_2\\\\ x(t) = C_1e^{3t}+3C_2e^{-2t}\\ y(t) = -C_1e^{3t}-8C_2e^{-2t}$