# Toronto Math Forum

## 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*}