64.23.7 problem 9
Internal
problem
ID
[13684]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
12,
Sturm-Liouville
problems.
Section
12.1,
Exercises
page
596
Problem
number
:
9
Date
solved
:
Tuesday, January 28, 2025 at 05:55:10 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\begin{align*} 2 x y^{\prime }+\left (x^{2}+1\right ) y^{\prime \prime }+\frac {\lambda y}{x^{2}+1}&=0 \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=0\\ y \left (1\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 5
dsolve([diff((x^2+1)*diff(y(x),x),x)+lambda/(x^2+1)*y(x)=0,y(0) = 0, y(1) = 0],y(x), singsol=all)
\[
y = 0
\]
✓ Solution by Mathematica
Time used: 1.126 (sec). Leaf size: 322
DSolve[{D[(x^2+1)*D[y[x],x],x]+\[Lambda]/(x^2+1)*y[x]==0,{y[0]==0,y[1]==0}},{y[x]},x,IncludeSingularSolutions -> True]
\[
y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} -\frac {\exp \left (\int _1^x\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right ) c_1 \left (\int _1^0\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]-\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]\right )}{\sqrt {x^2+1}} & \frac {\exp \left (\int _1^0\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}+\int _1^1\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right ) \int _1^1\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]}{\sqrt {2}}-\frac {\exp \left (\int _1^0\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}+\int _1^1\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right ) \int _1^0\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]}{\sqrt {2}}=0 \\ 0 & \text {True} \\ \end {array} \\ \end {array}
\]