8.5: Problems

8.1. Use the Method of Variation of Parameters to determine the general solution for the following problems.

a. \(y^{\prime \prime}+y=\tan x\).
b. \(y^{\prime \prime}-4 y^{\prime}+4 y=6 x e^{2 x}\)

8.2. Instead of assuming that \(c_{1}^{\prime} y_{1}+c_{2}^{\prime} y_{2}=0\) in the derivation of the solution using Variation of Parameters, assume that \(c_{1}^{\prime} y_{1}+c_{2}^{\prime} y_{2}=h(x)\) for an arbitrary function \(h(x)\) and show that one gets the same particular solution.

8.3. Find the solution of each initial value problem using the appropriate initial value Green's function.

a. \(y^{\prime \prime}-3 y^{\prime}+2 y=20 e^{-2 x}, \quad y(0)=0, \quad y^{\prime}(0)=6\).
b. \(y^{\prime \prime}+y=2 \sin 3 x, \quad y(0)=5, \quad y^{\prime}(0)=0\).
c. \(y^{\prime \prime}+y=1+2 \cos x, \quad y(0)=2, \quad y^{\prime}(0)=0\).
d. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=3 x^{2}-x, \quad y(1)=\pi, \quad y^{\prime}(1)=0\).

8.4. Consider the problem \(y^{\prime \prime}=\sin x, y^{\prime}(0)=0, y(\pi)=0\).

a. Solve by direct integration.
b. Determine the Green's function.
c. Solve the boundary value problem using the Green's function.
d. Change the boundary conditions to \(y^{\prime}(0)=5, y(\pi)=-3\).

i. Solve by direct integration.
ii. Solve using the Green's function.

8.5. Consider the problem:

\[\dfrac{\partial^{2} G}{\partial x^{2}}=\delta\left(x-x_{0}\right), \quad \dfrac{\partial G}{\partial x}\left(0, x_{0}\right)=0, \quad G\left(\pi, x_{0}\right)=0 \nonumber \]

a. Solve by direct integration.
b. Compare this result to the Green's function in part b of the last problem.
c. Verify that \(G\) is symmetric in its arguments.

8.6. In this problem you will show that the sequence of functions

\[f_{n}(x)=\dfrac{n}{\pi}\left(\dfrac{1}{1+n^{2} x^{2}}\right) \nonumber \]

approaches \(\delta(x)\) as \(n \rightarrow \infty\). Use the following to support your argument:

a. Show that \(\lim _{n \rightarrow \infty} f_{n}(x)=0\) for \(x \neq 0\).
b. Show that the area under each function is one.

8.7. Verify that the sequence of functions \(\left\{f_{n}(x)\right\}_{n=1}^{\infty}\), defined by \(f_{n}(x)= \dfrac{n}{2} e^{-n|x|}\), approaches a delta function.

8.8. Evaluate the following integrals:

a. \(\int_{0}^{\pi} \sin x \delta\left(x-\dfrac{\pi}{2}\right) d x\).
b. \(\int_{-\infty}^{\infty} \delta\left(\dfrac{x-5}{3} e^{2 x}\right)\left(3 x^{2}-7 x+2\right) d x\)
c. \(\int_{0}^{\pi} x^{2} \delta\left(x+\dfrac{\pi}{2}\right) d x\).
d. \(\int_{0}^{\infty} e^{-2 x} \delta\left(x^{2}-5 x+6\right) d x\). [See Problem 8.10.]
e. \(\int_{-\infty}^{\infty}\left(x^{2}-2 x+3\right) \delta\left(x^{2}-9\right) d x\). [See Problem 8.10.]

8.9. Find a Fourier series representation of the Dirac delta function, \(\delta(x)\), on \([-L, L]\)

8.10. For the case that a function has multiple simple roots, \(f\left(x_{i}\right)=0\), \(f^{\prime}\left(x_{i}\right) \neq 0, i=1,2, \ldots\), it can be shown that

\[\delta(f(x))=\sum_{i=1}^{n} \dfrac{\delta\left(x-x_{i}\right)}{\left|f^{\prime}\left(x_{i}\right)\right|} . \nonumber \]

Use this result to evaluate \(\int_{-\infty}^{\infty} \delta\left(x^{2}-5 x+6\right)\left(3 x^{2}-7 x+2\right) d x\).

8.11. Consider the boundary value problem: \(y^{\prime \prime}-y=x, x \in(0,1)\), with boundary conditions \(y(0)=y(1)=0\).

a. Find a closed form solution without using Green's functions.
b. Determine the closed form Green's function using the properties of Green's functions. Use this Green's function to obtain a solution of the boundary value problem.
c. Determine a series representation of the Green's function. Use this Green's function to obtain a solution of the boundary value problem.
d. Confirm that all of the solutions obtained give the same results.