3.7E: Exercises
- Page ID
- 17856
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In Exercises \((3.7E.1)\) to \((3.7E.40)\), find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients.
Exercise \(\PageIndex{1}\)
\(x^2y''-3xy'+(3+4x)y=0\)
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Exercise \(\PageIndex{2}\)
\(xy''+y=0\)
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Exercise \(\PageIndex{3}\)
\(4x^2(1+x)y''+4x(1+2x)y'-(1+3x)y=0\)
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Exercise \(\PageIndex{4}\)
\(xy''+xy'+y=0\)
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Exercise \(\PageIndex{5}\)
\(2x^2(2+3x)y''+x(4+21x)y'-(1-9x)y=0\)
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Exercise \(\PageIndex{6}\)
\(x^2y''+x(2+x)y'-(2-3x)y=0\)
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Exercise \(\PageIndex{7}\)
\(4x^2y''+4xy'-(9-x)y=0\)
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Exercise \(\PageIndex{8}\)
\(x^2y''+10xy'+(14+x)y=0\)
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Exercise \(\PageIndex{9}\)
\(4x^2(1+x)y''+4x(3+8x)y'-(5-49x)y=0\)
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Exercise \(\PageIndex{10}\)
\(x^2(1+x)y''-x(3+10x)y'+30xy=0\)
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Exercise \(\PageIndex{11}\)
\(x^2y''+x(1+x)y'-3(3+x)y=0\)
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Exercise \(\PageIndex{12}\)
\(x^2y''+x(1-2x)y'-(4+x)y=0\)
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Exercise \(\PageIndex{13}\)
\(x(1+x)y''-4y'-2y=0\)
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Exercise \(\PageIndex{14}\)
\(x^2(1+2x)y''+x(9+13x)y'+(7+5x)y=0\)
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Exercise \(\PageIndex{15}\)
\(4x^2y''-2x(4-x)y'-(7+5x)y=0\)
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Exercise \(\PageIndex{16}\)
\(3x^2(3+x)y''-x(15+x)y'-20y=0\)
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Exercise \(\PageIndex{17}\)
\(x^2(1+x)y''+x(1-10x)y'-(9-10x)y=0\)
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Exercise \(\PageIndex{18}\)
\(x^2(1+x)y''+3x^2y'-(6-x)y=0\)
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Exercise \(\PageIndex{19}\)
\(x^2(1+2x)y''-2x(3+14x)y'+(6+100x)y=0\)
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Exercise \(\PageIndex{20}\)
\(x^2(1+x)y''-x(6+11x)y'+(6+32x)y=0\)
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Exercise \(\PageIndex{21}\)
\(4x^2(1+x)y''+4x(1+4x)y'-(49+27x)y=0\)
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Exercise \(\PageIndex{22}\)
\(x^2(1+2x)y''-x(9+8x)y'-12xy=0\)
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Exercise \(\PageIndex{23}\)
\(x^2(1+x^2)y''-x(7-2x^2)y'+12y=0\)
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Exercise \(\PageIndex{24}\)
\(x^2y''-x(7-x^2)y'+12y=0\)
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Exercise \(\PageIndex{25}\)
\(xy''-5y'+xy=0\)
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Exercise \(\PageIndex{26}\)
\(x^2y''+x(1+2x^2)y'-(1-10x^2)y=0\)
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Exercise \(\PageIndex{27}\)
\(x^2y''-xy'-(3-x^2)y=0\)
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Exercise \(\PageIndex{28}\)
\(4x^2y''+2x(8+x^2)y'+(5+3x^2)y=0\)
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Exercise \(\PageIndex{29}\)
\(x^2y''+x(1+x^2)y'-(1-3x^2)y=0\)
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Exercise \(\PageIndex{30}\)
\(x^2y''+x(1-2x^2)y'-4(1+2x^2)y=0\)
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Exercise \(\PageIndex{31}\)
\(4x^2y''+8xy'-(35-x^2)y=0\)
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Exercise \(\PageIndex{32}\)
\(9x^2y''-3x(11+2x^2)y'+(13+10x^2)y=0\)
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Exercise \(\PageIndex{33}\)
\(x^2y''+x(1-2x^2)y'-4(1-x^2)y=0\)
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Exercise \(\PageIndex{34}\)
\(x^2y''+x(1-3x^2)y'-4(1-3x^2)y=0\)
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Exercise \(\PageIndex{35}\)
\(x^2(1+x^2)y''+x(5+11x^2)y'+24x^2y=0\)
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Exercise \(\PageIndex{36}\)
\(4x^2(1+x^2)y''+8xy'-(35-x^2)y=0\)
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Exercise \(\PageIndex{37}\)
\(x^2(1+x^2)y''-x(5-x^2)y'-(7+25x^2)y=0\)
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Exercise \(\PageIndex{38}\)
\(x^2(1+x^2)y''+x(5+2x^2)y'-21y=0\)
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Exercise \(\PageIndex{39}\)
\(x^2(1+2x^2)y''-x(3+x^2)y'-2x^2y=0\)
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Exercise \(\PageIndex{40}\)
\(4x^2(1+x^2)y''+4x(2+x^2)y'-(15+x^2)y=0\)
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Exercise \(\PageIndex{41}\)
(a) Under the assumptions of Theorem \((3.7.1)\), show that
\begin{eqnarray*}
y_1=x^{r_1}\sum_{n=0}^\infty a_n(r_1)x^n
\end{eqnarray*}
and
\begin{eqnarray*}
y_2=x^{r_2}\sum_{n=0}^{k-1}a_n(r_2)x^n+C\left(y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)
\end{eqnarray*}
are linearly independent.
Hint: Show that if \(c_1\) and \(c_2\) are constants such that \(c_1y_1+c_2y_2\equiv0\) on an interval \((0,\rho)\), then
\begin{eqnarray*}
x^{-r_2}(c_1y_1(x)+c_2y_2(x))=0,\quad 0<x<\rho.
\end{eqnarray*}
Then let \(x\to0+\) to conclude that \(c_2\)=0.
(b) Use the result of part (a) to complete the proof of Theorem \((3.7.1)\).
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Exercise \(\PageIndex{42}\)
Find a fundamental set of Frobenius solutions of Bessel's equation
\begin{eqnarray*}
x^2y''+xy'+(x^2-\nu^2)y=0
\end{eqnarray*}
in the case where \(\nu\) is a positive integer.
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Exercise \(\PageIndex{43}\)
Prove Theorem \((3.7.2)\).
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Exercise \(\PageIndex{44}\)
Under the assumptions of Theorem \((3.7.1)\), show that \(C=0\) if and only if \(p_1(r_2+ \ell)=0\) for some integer \(\ell\) in \(\{0,1,\dots,k-1\}\).
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Exercise \(\PageIndex{45}\)
Under the assumptions of Theorem \((3.7.2)\), show that \(C=0\) if and only if \(p_2(r_2+2\ell)=0\) for some integer \(\ell\) in \(\{0,1,\dots,k-1\}\).
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Exercise \(\PageIndex{46}\)
Let
\begin{eqnarray*}
Ly=\alpha_0x^2y''+\beta_0xy'+(\gamma_0+\gamma_1x)y
\end{eqnarray*}
and define
\begin{eqnarray*}
p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0.
\end{eqnarray*}
Show that if
\begin{eqnarray*}
p_0(r)=\alpha_0(r-r_1)(r-r_2)
\end{eqnarray*}
where \(r_1-r_2=k\), a positive integer, then \(Ly=0\) has the solutions
\begin{eqnarray*}
y_1=x^{r_1}\sum_{n=0}^\infty {(-1)^n\over n!\prod_{j=1}^n(j+k)}\left(\gamma_1\over\alpha_0\right)^n x^n
\end{eqnarray*}
and
\begin{eqnarray*}
y_2&=&x^{r_2}\sum_{n=0}^{k-1} {(-1)^n\over n!\prod_{j=1}^n(j-k)} \left(\gamma_1\over\alpha_0\right)^n x^n\\
&&-{1\over k!(k-1)!}\left(\gamma_1\over\alpha_0\right)^k\left(y_1\ln x- x^{r_1}\sum_{n=1}^\infty {(-1)^n\over n!\prod_{j=1}^n(j+k)}\left(\gamma_1\over\alpha_0\right)^n \left(\sum_{j=1}^n{2j+k\over j(j+k)}\right)x^n\right).
\end{eqnarray*}
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Exercise \(\PageIndex{47}\)
Let
\begin{eqnarray*}
Ly=\alpha_0x^2y''+\beta_0xy'+(\gamma_0+\gamma_2x^2)y
\end{eqnarray*}
and define
\begin{eqnarray*}
p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0.
\end{eqnarray*}
Show that if
\begin{eqnarray*}
p_0(r)=\alpha_0(r-r_1)(r-r_2)
\end{eqnarray*}
where \(r_1-r_2=2k\), an even positive integer, then \(Ly=0\) has the solutions
\begin{eqnarray*}
y_1=x^{r_1}\sum_{m=0}^\infty {(-1)^m\over 4^mm!\prod_{j=1}^m(j+k)}\left(\gamma_2\over\alpha_0\right)^m x^{2m}
\end{eqnarray*}
and
\begin{eqnarray*}
y_2&=&x^{r_2}\sum_{m=0}^{k-1} {(-1)^m\over4^mm!\prod_{j=1}^m(j-k)} \left(\gamma_2\over\alpha_0\right)^m x^{2m}\\
&&-{2\over 4^kk!(k-1)!}\left(\gamma_2\over\alpha_0\right)^k\left(y_1\ln x- {x^{r_1}\over2}\sum_{m=1}^\infty {(-1)^m\over 4^mm!\prod_{j=1}^m(j+k)}\left(\gamma_2\over\alpha_0\right)^m \left(\sum_{j=1}^m{2j+k\over j(j+k)}\right)x^{2m}\right).
\end{eqnarray*}
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Exercise \(\PageIndex{48}\)
Let \(L\) be as in Exercises \((3.5E.57)\) and \((3.5E.58)\), and suppose the indicial polynomial of \(Ly=0\) is
\begin{eqnarray*}
p_0(r)=\alpha_0(r-r_1)(r-r_2),
\end{eqnarray*}
with \(k=r_1-r_2\), where \(k\) is a positive integer. Define \(a_0(r)=1\) for all \(r\). If \(r\) is a real number such that \(p_0(n+r)\)
is nonzero for all positive integers \(n\), define
\begin{eqnarray*}
a_n(r)=-{1\over p_0(n+r)}\sum_{j=1}^n p_j(n+r-j)a_{n-j}(r),\,n\ge1,
\end{eqnarray*}
and let
\begin{eqnarray*}
y_1=x^{r_1}\sum_{n=0}^\infty a_n(r_1)x^n.
\end{eqnarray*}
Define
\begin{eqnarray*}
a_n(r_2)=-{1\over p_0(n+r_2)}\sum_{j=1}^n p_j(n+r_2-j)a_{n-j}(r_2)\mbox{ if } n\ge1\mbox{ and }n\ne k,
\end{eqnarray*}
and let \(a_k(r_2)\) be arbitrary.
(a) Conclude from Exercise \((3.6E.66)\) that
\begin{eqnarray*}
L\left(y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)=k\alpha_0x^{r_1}.
\end{eqnarray*}
(b) Conclude from Exercise \((3.5E.57)\) that
\begin{eqnarray*}
L\left(x^{r_2}\sum_{n=0}^\infty a_n(r_2)x^n\right)=Ax^{r_1},
\end{eqnarray*}
where
\begin{eqnarray*}
A=\sum_{j=1}^k p_j(r_1-j)a_{k-j}(r_2).
\end{eqnarray*}
(c) Show that \(y_1\) and
\begin{eqnarray*}
y_2=x^{r_2}\sum_{n=0}^\infty a_n(r_2)x^n -{A\over k\alpha_0} \left(y_1 \ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)
\end{eqnarray*}
form a fundamental set of Frobenius solutions of \(Ly=0\).
(d) Show that choosing the arbitrary quantity \(a_k(r_2)\) to be nonzero merely adds a multiple of \(y_1\) to \(y_2\). Conclude that we may as well take \(a_k(r_2)=0\).
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