MAT244--2019F > Quiz-5

LEC0101 quiz5

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Linqian Shen:
Find the general solution of the given differential equation.
$$y^{\prime \prime}+y=\tan (t)$$

$$\begin{array}{c}{r^{2}+1=0} \\ {r=\pm i} \\ {y=c_{1} \cos t+c_{2} \sin t}\end{array}$$
\begin{aligned} w&=\left|\begin{array}{cc}{\cos t} & {\sin t} \\ {-\sin t} & {\cos t}\end{array}\right|\\ &=\cos ^{2} t+\sin ^{2} t=1\\ w_{1}&=\left|\begin{array}{ll}{0} & {\sin t} \\ {1} & {\cos t}\end{array}\right|=-\sin t\\ w_{2}&=\left|\begin{array}{cc}{\cos t} & {0} \\ {-\sin t} & {1}\end{array}\right|=\cos t \end{aligned}
\begin{aligned} y p(t)&=\cos t \int \frac{-s i n t-\tan s}{1} d s+s \sin t \int \frac{\cos s-t a n s}{1} d s\\ &=\cos t \int-\sin s \frac{\sin s}{\cos s} d s+\sin t \int \cos s \frac{\sin s}{\cos s} d s\\ &=-\cos t \int \frac{1-\cos ^{2} s}{\cos s} d s+\sin t \int \sin s d s\\ &=-\cos t \int \sec -\cos s d s+\sin t(-\cos s)\\ &=-\cos \ln (\sec t+\tan t)+\cos t \sin t-\sin t \cos t \end{aligned}
$$y(t)=c_{1} \cos t+c_{2} \sin t-\cos t \ln (\sec t+\tan t)$$