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\textbf{Math 365: Honors Analysis I}
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\textbf{Mid-Term Exam}\quad \textsl{October 13, 2000}

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\emph{There are eight problems, each worth 15 points}

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1. Find the general solution of $y' + \tan(x) y = \cos(x)$.

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2. Find the general solution of $y''' + y'' - 2y = x^5$.

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3. Find two independent solutions of
\[ y'' - x^2 y' - x y = 0 \]

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4. Let $f(x) = x^2$.

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a) Find the Fourier series of $f(x)$ on $[-\pi,\pi]$.

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b) Find the sine series of $f(x)$ on $[0,1]$.

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5. Show that any solution of Bessel's Equation
\[ x^2 y'' + x y' + (x^2 - p^2) y = 0 \]
has infinitely positive zeros.

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6. Describe the Frobenius series solutions for a second order linear
differential equation and the number of such solutions that exist.

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7. Let $P(x)$ and $Q(x)$ be analytic at 0 with radii of convergence~$\ge
R$. Prove that any power series solution of
\[ y'' + P(x)y' + Q(x) y = 0 \]
has a radius of convergence~$\ge R$.

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8. Prove that if $f(x)$ is a piecewise smooth function on $[-\pi,\pi]$, is
periodic with period $2\pi$, and is defined at points of discontinuity by
\[ \frac{1}{2}[f(x^-) + f(x^+)] \]
then the Fourier series of $f(x)$ converges to $f(x)$ for all~$x$.

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