LEC5101 Quiz5

[Wang Jingyao]

Find the general solution of the given differential equation.

$y’’+4y=3\csc(2t)$,$\quad 0<t<\dfrac{\pi}{2}$

For homogeneous equation:

$y’’+4y=0$

Characteristic equation:

$r^2+4=0$ $\Longrightarrow$ $\left\{\begin{array}{l}r_1=2i\\r_2=-2i \end{array}\right.$

Complementary solution:

$y_c(t)=c_1\cos(2t)+c_2\sin(2t)$

For nonhomogeneous equation $ y’’+4y=3\csc(2t)$, we have:

$p(t)=0,\quad q(t)=4,\quad g(t)=3\csc(2t)$

Then,

$W[y_1(t),y_2(t)]=$$
\left |
\begin{matrix}
y_1(t) & y_2(t) \\
y_1’(t) & y_2’(t)
\end{matrix}
\right |
$$
=$$
\left |
\begin{matrix}
\cos(2t) & \sin(2t) \\
-2\sin(2t) & 2\cos(2t)
\end{matrix}
\right |
$$
=2$

Thus, $y_1(t)$ and $y_2(t)$ form a fundamental set of solutions. Therefore,

$$
\begin{align}
u_1(t)&=-\int\dfrac{y_2(t)g(t)}{ W[y_1(t),y_2(t)]}dt\\
\notag \\
&=-\int\dfrac{\sin(2t) \cdot 3\csc(2t)}{2}dt\\
\notag \\
&=-\int\dfrac{3}{2}dt\\
\notag \\
&=-\dfrac{3}{2}t\\
\end{align}
$$

$$
\begin{align}
u_2(t)&=\int\dfrac{y_1(t)g(t)}{ W[y_1(t),y_2(t)]}dt\\
\notag \\
&=\int\dfrac{\cos(2t) \cdot 3\csc(2t)}{2}dt\\
\notag \\
&=\dfrac{3}{2}\int\dfrac{\cos(2t)}{\sin(2t)}dt\\
\notag \\
&=\dfrac{3}{2}\int \cot(2t)dt\\
\notag \\
&=\dfrac{3}{4}\ln|\sin(2t)|\\
\end{align}
$$

Since,

$Y(t)=u_1(t)y_1(t)+u_2(t)y_2(t)$

Therefore,

$$
\begin{align}
Y(t)&= \cos(2t) \cdot (-\dfrac{3}{2}t)+\sin(2t) \cdot (\dfrac{3}{4}\ln|\sin(2t)|) \notag \\
\notag \\
&=\dfrac{3}{4}\sin(2t)\ln|\sin(2t)|-\dfrac{3}{2}t\cos(2t) \notag \\
\end{align}
$$

Thus, the general solution is,

$$
\begin{align}
y(t)&=y_c(t)+Y(t) \notag \\
\notag \\
y(t)&= c_1\cos(2t)+c_2\sin(2t)+ \dfrac{3}{4}\sin(2t)\ln|\sin(2t)|-\dfrac{3}{2}t\cos(2t) \notag \\
\end{align}
$$