64.23.8 problem 10
Internal
problem
ID
[13685]
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
:
10
Date
solved
:
Tuesday, January 28, 2025 at 05:55:13 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\begin{align*} -\frac {6 y^{\prime } x}{\left (3 x^{2}+1\right )^{2}}+\frac {y^{\prime \prime }}{3 x^{2}+1}+\lambda \left (3 x^{2}+1\right ) y&=0 \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=0\\ y \left (\pi \right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.030 (sec). Leaf size: 5
dsolve([diff(1/(3*x^2+1)*diff(y(x),x),x)+lambda*(3*x^2+1)*y(x)=0,y(0) = 0, y(Pi) = 0],y(x), singsol=all)
\[
y = 0
\]
✓ Solution by Mathematica
Time used: 0.323 (sec). Leaf size: 450
DSolve[{D[1/(3*x^2+1)*D[y[x],x],x]+\[Lambda]*(3*x^2+1)*y[x]==0,{y[0]==0,y[1]==0}},{y[x]},x,IncludeSingularSolutions -> True]
\[
y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} -\exp \left (\int _1^x-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right ) \sqrt {3 x^2+1} c_1 \left (\int _1^0\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]-\int _1^x\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]\right ) & 2 \exp \left (\int _1^0-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}+\int _1^1-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right ) \int _1^1\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]-2 \exp \left (\int _1^0-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}+\int _1^1-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right ) \int _1^0\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]=0 \\ 0 & \text {True} \\ \end {array} \\ \end {array}
\]