Решить cистему dx\dt=6*x+3*y dy\dt=-8*x-5*y
Ну в чем проблема, найди собственные значения и вперед)
а помочь никак
пожалуйста
А, ладно, так и быть
Найдем базисные решения в виде Нашли собственные значения, теперь собственные вектора
ничего не пойму
какие - то каракули
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![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](https://tex.z-dn.net/?f=x+%3D+Ae%5E%7B%5Clambda+t%7D%5Cquad+y%3DBe%5E%7B%5Clambda+t%7D%5C%5C%5C%5C%0A%5Cleft%5C%7B%0A%5Cbegin%7Baligned%7D%0A%26%5Clambda+A+%3D+6A%2B3B%5C%5C%0A%26%5Clambda+B+%3D+-8A-5B%0A%5Cend%7Baligned%7D%0A%5Cright.%5C%5C%5C%5C%5C%5C%0A+%5Cdet+%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D6-%5Clambda%263%5C%5C-8%26-5-%5Clambda%5Cend%7Barray%7D%5Cright%5D+%3D0%5C%5C%5C%5C%0A%28%5Clambda%2B5%29%28%5Clambda-6%29%2B24+%3D+0%5C%5C%0A%5Clambda%5E2+-%5Clambda-6+%3D+0%5C%5C%0A%5Clambda_1+%3D+3%5C%5C%0A%5Clambda_2+%3D+-2%0A "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")
![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}](https://tex.z-dn.net/?f=1%29%5C%5C%0A%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D3%263%5C%5C-8%26-8%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7DA%5C%5CB%5Cend%7Barray%7D%5Cright%5D+%3D0%5C%5C%5C%5C%0AA+%3D+-B+%3D+C_1%0A%0A2%29%5C%5C%0A%5Cleft%5B%5Cbegin%7Barray%7D%7Bcc%7D8%263%5C%5C-8%26-3%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7DA%5C%5CB%5Cend%7Barray%7D%5Cright%5D+%3D0%5C%5C%5C%5C%0AA+%3D+3C_2%5Cquad+B%3D-8C_2%5C%5C%5C%5C%0Ax%28t%29+%3D+C_1e%5E%7B3t%7D%2B3C_2e%5E%7B-2t%7D%5C%5C%0Ay%28t%29+%3D+-C_1e%5E%7B3t%7D-8C_2e%5E%7B-2t%7D "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}")