Author Topic: TUT 5103 Quiz4  (Read 489 times)

Zuwei Zhao

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TUT 5103 Quiz4
« on: October 18, 2019, 02:00:02 PM »
$$
\begin{array}{l}{y^{\prime \prime}+y=0, \quad y\left(\frac{\pi}{3}\right)=2, \quad y^{\prime}\left(\frac{\pi}{3}\right)=-4} \\ {r^{2}+1=0}\end{array}
$$
$$
\begin{aligned} r &=\frac{-0 \pm \sqrt{0-4 \cdot 1} \cdot 1}{2} \\ &=\frac{-0+2 i}{2} \\ &=0 \pm i \end{aligned}
$$
$$
\begin{array}{l}{y=c_{1} e^{o t} \cos t+c_{2} e^{o t} \sin t} \\ {y=c_{1} \cos t+c_{2} \sin t} \\ {y^{\prime}=-c_{1} \sin t+c_{2} \cos t}\end{array}
$$
$$
\left\{\begin{array}{l}{2=\frac{1}{2} c_{1}+\frac{\sqrt{3}}{2} c_{2}} \\ {-4=-\frac{\sqrt{3}}{2} c_{1}+\frac{1}{2} c_{2}}\end{array}\right.
$$
$$
\begin{aligned} c_{1} &=1+2 \sqrt{3}\quad c_{2}=\sqrt{3}-2 \\ \therefore y &=(1+2 \sqrt{3}) \cos t+(\sqrt{3}-2) \sin t \end{aligned}
$$