# Toronto Math Forum

## MAT244--2019F => MAT244--Test & Quizzes => Quiz-5 => Topic started by: Ruodan Chen on November 01, 2019, 03:41:59 PM

Title: TUT5103 Quiz5
Post by: Ruodan Chen on November 01, 2019, 03:41:59 PM
Find the general solution of the given differential equation.

y'' + 4y = 3csc(2t), 0 < t < $\frac{\pi}{2}$

$r^{2}$ + 4 = 0

$r^{2}$ = $-4$

r = $\pm2i$

$y_{c}$(t) = $c_{1}$cos2t + $c_{2}$sin2t

$y_{1}$ = cos2t

$y_{2}$ = sin2t

w = $\begin{array}{cc} cos2t & sin2t\\ -2sin2t & 2cos2t \end{array}$ = 2$cos^{2}2t$ + 2$sin^{2}2t$ = 2

$w_{1}$ = $\begin{array}{cc} 0 & sin2t\\ 1 & 2cos2t \end{array}$ = -sin2t

$w_{2}$ = $\begin{array}{cc} cos2t & 0\\ -2sin2t & 1 \end{array}$ = cos2t

Y(t) = cos2t$\int$$\frac{-sin2s3csc2s}{2}ds + sin\int\frac{cos2s3csc2s}{2}ds = -cos2t(\frac{3}{2})\intsin2scsc2sds + sin2t(\frac{3}{2})\int$$\frac{cos2s}{sin2s}$ds

= -cos2t($\frac{3}{2}$)t + sin2t($\frac{3}{2}$)ln|sin2t|

y(t) = $y_{c}$(t) + Y(t) = $c_{1}$cos2t + $c_{2}$sin2t - cos2t($\frac{3}{2}$)t + sin2t($\frac{3}{2}$)ln|sin2t|