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Exercises 4.13 Exercise
1.
Solve the following differential equations.
\begin{equation*}
\frac{d^{2}y}{dx^{2}}+y= \mathrm{cosec}x
\end{equation*}
Answer .
\begin{equation*}
C_{1}\cos x+C_{2}\sin x -x\cos x+\sin x\cdot \log\sin x.
\end{equation*}
\begin{equation*}
\frac{d^{2}y}{dx^{2}}+4y=4\tan 2x, y(0)=0,y(\frac{\pi}{6})=0.
\end{equation*}
Answer .
\begin{equation*}
\frac{1}{\sqrt{3}}\log(2+\sqrt{3})\sin 2x-\cos 2x\log(sec 2x+\tan 2x).
\end{equation*}
\begin{equation*}
\frac{d^{2}y}{dx^{2}}-2\frac{dy}{dx}+2y=e^{x}\tan x.
\end{equation*}
Answer .
\begin{equation*}
e^{x}(C_{1}\cos x+C_{2}\sin x)-e^{x}\cos x\sin x(\sec x+\tan x).
\end{equation*}
2.
Solve
\begin{equation*}
x^{2}y''-2x(1+x)y' +2(1+x)y =x^{3}.
\end{equation*}
Answer .
\begin{equation*}
y=(C_{1}x+C_{2}xe^{x})-(\frac{1}{2}x^{2}+\frac{1}{4}x).
\end{equation*}
3.
Solve
\begin{equation*}
(1-x)y''+xy' -y =(1-x^{2}).
\end{equation*}
Answer .
\begin{equation*}
y=(C_{1}x+C_{2}e^{x})+(1+x+x^{2}).
\end{equation*}
4.
Solve
\begin{equation*}
y''+(1-\cot x)y' -y\cot x =\sin^{2}x.
\end{equation*}
Answer .
\begin{equation*}
y=\{C_{1}(\sin x-\cos x)+C_{2}e^{-x}\}-\frac{1}{10}(\sin 2x)-2\cos 2x).
\end{equation*}
5. Apply power series method to find the solution of the following linear equations.
\begin{equation*}
y''-(x+1)y'+x^{2}y =x.
\end{equation*}
With the initial conditions \(y(0) =1, y'(0)=1\text{.}\)
Answer .
\begin{equation*}
y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{2}+\frac{x^{4}}{8}+\cdots
\end{equation*}
\begin{equation*}
y''+x^{2}y =0.
\end{equation*}
Answer .
\begin{equation*}
y=a_{o}\left[1-\frac{x^{4}}{12}+\frac{x^{8}}{12\cdot 7\cdot 8}-\frac{x^{12}}{12\cdot 8\cdot 7 \cdot 11 \cdot 12} +\cdots\right]
\end{equation*}
\begin{equation*}
+a_{1}\left[x-\frac{x^{2}}{20}+\frac{x^{9}}{20\cdot 8\cdot 9}-\cdots\right].
\end{equation*}
\begin{equation*}
xy''+2y'+xy =0.
\end{equation*}
Answer .
\begin{equation*}
y=c_{1}\left[1-\frac{x^{2}}{3!}+\frac{x^{4}}{5!}-\cdots\right]+c_{2}x^{-1}\left[1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\cdots\right]
\end{equation*}
\begin{equation*}
xy''+(x-1)y'-y=0.
\end{equation*}
Answer .
\begin{equation*}
y=A\left[1-\frac{x}{1!}+\frac{x^{2}}{2!}-\frac{x^{3}}{3!}+\cdots\right]
\end{equation*}
\begin{equation*}
+Bx^{2}\left[1-\frac{2x}{3!}+\frac{2x^{2}}{4!}-\frac{2x^{3}}{5!}+\cdots\right].
\end{equation*}
\begin{equation*}
2x(1-x)y''+(1-x)y'+3y=0.
\end{equation*}
Answer .
\begin{equation*}
y=a_{o}\sqrt{x}(1-x)+a_{1}\left[1-3x+\frac{3x^{2}}{1\cdot 3}+\frac{3x^{3}}{3\cdot 5}+\cdots\right].
\end{equation*}
6.
Express in terms of Legendre’s Polynomials.
\begin{equation*}
x^{4}+x^{3}+x^{2}+x+1
\end{equation*}
Answer .
\begin{equation*}
\frac{8}{35}P_{4}(x)+ \frac{2}{5}P_{3}(x)+\frac{26}{21}P_{2}(x)+\frac{3}{5}P_{1}(x)+\frac{23}{15}P_{0}(x)
\end{equation*}
\begin{equation*}
1+2x-3x^{2}+4x^{3}.
\end{equation*}
Answer .
\begin{equation*}
\frac{8}{5}P_{3}(x)-2P_{2}(x)+\frac{22}{5}P_{1}(x)+\frac{3}{5}P_{1}(x).
\end{equation*}
7.
From the values of Legendre’s Polynomials, prove that
\begin{equation*}
x^{2}=\frac{1}{3}[2 P_{2}(x)+P_{0}(x)]
\end{equation*}
\begin{equation*}
x^{5}=\frac{1}{63}[8 P_{5}(x)+28 P_{3}(x)+27P_{1}(x)].
\end{equation*}
8.
Prove that
\begin{equation*}
\int\limits_{-1}^{+1}P_{n}(x)\,dx =0, \quad n \neq 0.
\end{equation*}
9.
Prove that
\begin{equation*}
\int\limits_{-1}^{+1} x P_{n}(x)P_{n-1}(x)\,dx =\frac{2n}{4n^{2}-1}.
\end{equation*}
10.
Show that
\begin{equation*}
x^{2r}=\sum \limits_{n=0}^{r}\frac{2^{2n}(4n+1)2r!(n-r)!}{(2r+2n+1)!(r-n)!}P_{2n}(x).
\end{equation*}
11.
Prove that
\begin{equation*}
P_{n}(0) =0
\end{equation*}
for \(n\) odd.
12. Find the value of \(J_{\pm\frac{5}{2}}(x)\text{.}\)
Answer .
\begin{equation*}
J_{+\frac{5}{2}}(x)=\sqrt{\frac{2}{\pi x}}\left[\frac{3-x^{2}}{x^{2}}\sin x-\frac{3}{x}\cos x\right]
\end{equation*}
and
\begin{equation*}
J_{-\frac{5}{2}}(x)=\sqrt{\frac{2}{\pi x}}\left[\frac{3}{x}\sin x-\frac{3-x^{2}}{x^{2}}\cos x\right]
\end{equation*}
13.
Prove that
\begin{equation*}
J_{4}(x)=\left(\frac{48}{x^{3}}-\frac{8}{x}\right)J_{1}(x)+\left(1-\frac{24}{x^{2}}\right)J_{o}(x)
\end{equation*}
14.
Prove that
\begin{equation*}
J_{1}(x)=\frac{x}{2}-\frac{x^{3}}{2^{3}1!2!}+\frac{x^{5}}{2^{5}2!3!}-\frac{x^{7}}{2^{7}3!4!}+\cdots
\end{equation*}
15.
Show that
\begin{equation*}
J''_{o}(x)=\frac{1}{2}\left[J_{2}(x)-J_{o}(x)\right]
\end{equation*}
16.
Prove that
\begin{equation*}
\frac{\,d}{\,dx}\left[xJ_{1}(x)\right]=xJ_{o}(x)
\end{equation*}
17.
Prove that
\begin{equation*}
4J''_{o}(x)+3J_{o}(x)+J_{3}(x)=0
\end{equation*}
18.
Prove that
\begin{equation*}
J^{2}_{o}+2J^{2}_{1}+2J^{2}_{2}+\cdots=1
\end{equation*}
19.
Prove that
\begin{equation*}
\int\limits_{-\infty}^{\infty}x^{2}e^{-x^{2}}H_{m}(x)H_{n}(x)\,dx =\sqrt{\pi}2^{n}n!(n+1)/2\cdot\delta_{m,n}.
\end{equation*}
20.
Show that
\begin{equation*}
\int\limits_{-\infty}^{\infty}xH_{n}(x)e^{-x^{2}}\,dx
=\begin{cases}
0, \\
2\pi.
\end{cases}
\end{equation*}
21.
Show that
\begin{equation*}
x^{2r}=\frac{(2r)!}{2^{2r}}\sum\limits_{n=0}^{r}\frac{H_{2n}(x)}{(2r)!(r-n)!}.
\end{equation*}
22.
Show that
\begin{equation*}
H_{1}(x)=\begin{cases}
2xH_{o}(x)\quad \text{for n even }\\
\frac{(n+1)!}{(n+1)/2!} \quad \text{for n odd }.
\end{cases}
\end{equation*}
23.
Prove that
\begin{equation*}
L_{n}(x)= \sum\limits_{r=0}^{\infty}\frac{(-1)^{r}}{r!}\comb{n}{r}x^{r}.
\end{equation*}
24.
Find the values of
\begin{equation*}
\int\limits_{0}^{\infty}e^{-x}L_{2}(x)L_{5}(x)\,dx
\end{equation*}
Answer .
\begin{equation*}
0.
\end{equation*}
\begin{equation*}
\int\limits_{0}^{\infty}e^{-x}L^{2}_{4}(x)\,dx
\end{equation*}
Answer .
\begin{equation*}
1.
\end{equation*}
25.
Prove that
\begin{equation*}
L'_{n}(x)=xL'_{n-1}+nL_{n-1}(x) =0.
\end{equation*}
