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\problemgrouptitle{Differential Equations}

\item Show $y=x^3+2x+2$ is a solution of the differential equation $2y''+3xy'-9y+18=0$\,.

\item Show $y(t)=A\cos\left(2t)+B\sin(2t\right)$ is a general solution to the differential equation \mbox{$y''+4y=0$}\,.  If as well $y(0)=3$ and $y'(0)=5$, find the constants $A$ and $B$ for the particular solution to that initial value problem.
\answer{$\displaystyle y=3\cos(2t)+\frac{5}{2}\sin(2t)$}

\begin{problemblock}{Solve the following differential equations and initial value problems.}

\item $\displaystyle \frac{dy}{dx}=\sqrt{x}+\csc x\cot x$
\answer{$\displaystyle y=\frac{2}{3}x^{\frac{3}{2}}-\csc x+C$}

\item $\displaystyle \frac{d^2x}{dt^2}=(2t+3)^5$
\answer{$\displaystyle x(t)=\frac{1}{168}(2t+3)^7+Ct+D$}

\item $\displaystyle y'=2x+\sin x$, $y(0)=1$
\answer{$\displaystyle y=x^2-\cos x+2$}

\end{problemblock}
