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MAT244--2018F => MAT244--Lectures & Home Assignments => Topic started by: wenlinwang on November 27, 2018, 05:53:14 PM

Title: final review : matrix of non-home variation of parameter
Post by: wenlinwang on November 27, 2018, 05:53:14 PM
Find the general solution of the system of ODEs
$x_t' = x + y + \frac{e^t}{\cos (t)}$
$y_t' = -x + y + \frac{e^t}{\sin(t)}$
Title: Re: final review : matrix of non-home variation of parameter
Post by: Meiyi Lu on November 27, 2018, 06:02:47 PM
\begin{align*}
    \begin{bmatrix}
    x' \\ y'
    \end{bmatrix} &=
    \begin{bmatrix}
    1 & 1 \\
    -1 & -1
    \end{bmatrix}
    \begin{bmatrix}
    x \\ y
    \end{bmatrix}
    +
    \begin{bmatrix}
        \frac{e^t}{\cos t}\\
        \frac{e^t}{\sin t}
    \end{bmatrix}
\end{align*}
\begin{align*}
    x(t) &= e^{(1+i)t}
    \begin{bmatrix}
    1 \\ i
    \end{bmatrix}\\&= e^t
    \begin{bmatrix}
    \cos t + i\sin t\\
    i\cos t - \sin t
    \end{bmatrix}\\
    &= e^t
    \begin{bmatrix}
    \cos t\\
    -\sin t
    \end{bmatrix} + ie^i
    \begin{bmatrix}
    \sin t\\
    \cos t
    \end{bmatrix}
\end{align*}
\begin{equation*}
    \phi = \begin{bmatrix}
    e^t \cos t & e^t \sin t \\
    -e^t \sin t & e^t \cos t
    \end{bmatrix}
\end{equation*}
\begin{equation*}
    \phi u' = g
\end{equation*}
\begin{equation*}
    u_1' = 0
\end{equation*}
\begin{equation*}
    u_2' = \frac{1}{\sin t \cos t }
\end{equation*}
\begin{equation*}
    u_2 = \int \frac{1}{\sin t \cos t} dt = \ln |\tan t| + c_2
    \end{equation*}
\begin{equation*}
    u_1 = c_1
\end{equation*}
\begin{equation*}
    x = \phi u = c_1
    \begin{bmatrix}
    e^t \cos t \\ -e^t \sin t
    \end{bmatrix} + (c_2 + \ln |\tan t|)
    \begin{bmatrix}
    e^t\sin t \\
    e^t\cos t 
    \end{bmatrix}
\end{equation*}