Question

The coefficient matrix for a system of linear differential equations of the form y_1=Ay has the given eigenvalues and eigenspace bases. Find the gener

Forms of linear equations
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asked 2020-11-02
The coefficient matrix for a system of linear differential equations of the form \(y_1=Ay\)
has the given eigenvalues and eigenspace bases. Find the general solution for the system
\(\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}\)

Answers (1)

2020-11-03

The general solution is
\(y=c_1y_1+c_2y_2\)
\(\text{for } y_1=e^{at}(\cos(bt)Re(u)-\sin(bt)Im(u))\)
\(y_2=e^{at}(\sin(bt)Re(u)+\cos(bt)Im(u))\)
We have
\(\lambda_1=3+i \Rightarrow \left\{ \begin{bmatrix}2i\\ i \end{bmatrix} \right\} , \lambda_2=3-i \Rightarrow \left\{ \begin{bmatrix}-2i\\ -i \end{bmatrix} \right\}\)
Then
\(y_1=e^{3t}\bigg(\cos(t)\begin{bmatrix}0\\0\end{bmatrix}-\sin(t)\begin{bmatrix}2\\1\end{bmatrix}\bigg)\)
\(y_2=e^{t}\bigg(\sin(-t)\begin{bmatrix}0\\0\end{bmatrix}+\cos(-t)\begin{bmatrix}-2\\-1\end{bmatrix}\bigg)\)
Hence the general solution is
\(y=c_1y_1+c_2y_2\)
\(=c_1e^{3t}\bigg(\cos(t)\begin{bmatrix}0\\0\end{bmatrix}-\sin(t)\begin{bmatrix}2\\1\end{bmatrix}\bigg)+c_2e^t\bigg(\sin(-t)\begin{bmatrix}0\\0\end{bmatrix}+\cos(-t)\begin{bmatrix}-2\\-1\end{bmatrix}\bigg)\)
The individual functions are
\(y_1=-2c_1e^{3t}(\sin(t)+\cos(-t))\)
\(y_2=-c_2e^{3t}(\sin(t)+\cos(-t))\)

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