64.23.7 problem 9
Internal
problem
ID
[13613]
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
:
Monday, March 31, 2025 at 08:02:45 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*}
✓ Maple. Time used: 0.018 (sec). Leaf size: 5
ode:=2*x*diff(y(x),x)+(x^2+1)*diff(diff(y(x),x),x)+lambda/(x^2+1)*y(x) = 0;
ic:=y(0) = 0, y(1) = 0;
dsolve([ode,ic],y(x), singsol=all);
\[
y = 0
\]
✓ Mathematica. Time used: 1.126 (sec). Leaf size: 322
ode=D[(x^2+1)*D[y[x],x],x]+\[Lambda]/(x^2+1)*y[x]==0;
ic={y[0]==0,y[1]==0};
DSolve[{ode,ic},{y[x]},x,IncludeSingularSolutions->True]
\[
y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} -\frac {\exp \left (\int _1^x\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right ) c_1 \left (\int _1^0\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]-\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]\right )}{\sqrt {x^2+1}} & \frac {\exp \left (\int _1^0\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}+\int _1^1\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right ) \int _1^1\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]}{\sqrt {2}}-\frac {\exp \left (\int _1^0\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}+\int _1^1\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right ) \int _1^0\exp \left (-2 \int _1^{K[2]}\frac {\unicode {f80d}+i \sqrt {\lambda }}{\unicode {f80d}^2+1}d\unicode {f80d}\right )dK[2]}{\sqrt {2}}=0 \\ 0 & \text {True} \\ \end {array} \\ \end {array}
\]
✗ Sympy
from sympy import *
x = symbols("x")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(lambda_*y(x)/(x**2 + 1) + 2*x*Derivative(y(x), x) + (x**2 + 1)*Derivative(y(x), (x, 2)),0)
ics = {y(0): 0, y(1): 0}
dsolve(ode,func=y(x),ics=ics)
ValueError : Couldnt solve for initial conditions