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\begin{center}
{\Large Warm-Up Problems and Lecture Problems

February 24, 2003}
\end{center}

\begin{enumerate}


\item  Using Euler's method, with step size $0.5$, estimate $y(3)$ for the initial value problem:
	\[
	\frac{dy}{dx} = y-xy \qquad y(2) = 4
	\]

\item   Consider the differential equation:
	\[
	\frac{dy}{dx} = \frac{x^7-12\sin^5x}{x^4+1}
	\]
	Is the following function a solution to this differential equation?  Why or why not?
	\[
	F(x) = \int_0^x \frac{t^7-12\sin^5t}{t^4+1} \; dt
	\]

\newpage 

\lecture

\item  Determine if the following differential equations are separable.
	After making this determination for all the equations, solve the separable equations.
	\begin{enumerate}
	\item  $\frac{dy}{dx} = 2x\sqrt{y}$
	\item  $\frac{dy}{dx} - 2y = 3e^{2x}$
	\item  $x\frac{dy}{dx} +3y = 2x^5 \quad y(2)=1$
	\item  $\frac{dy}{dx} = \frac{1}{4y^3} \quad y(0)=1$
	\item  $\frac{dy}{dx} = 3x^2y^2-y^2 \quad y(0)=1$
	\item  $\frac{dy}{dx} + 2xy = x$
	\end{enumerate}


\item  Find a family of curves orthogonal to the family $y=\frac{k}{x}$.

\end{enumerate}



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