64.23.8 problem 10
Internal
problem
ID
[13614]
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
:
Monday, March 31, 2025 at 08:02:47 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*}
✓ Maple. Time used: 0.017 (sec). Leaf size: 5
ode:=-6/(3*x^2+1)^2*diff(y(x),x)*x+1/(3*x^2+1)*diff(diff(y(x),x),x)+lambda*(3*x^2+1)*y(x) = 0;
ic:=y(0) = 0, y(Pi) = 0;
dsolve([ode,ic],y(x), singsol=all);
\[
y = 0
\]
✓ Mathematica. Time used: 0.323 (sec). Leaf size: 450
ode=D[1/(3*x^2+1)*D[y[x],x],x]+\[Lambda]*(3*x^2+1)*y[x]==0;
ic={y[0]==0,y[1]==0};
DSolve[{ode,ic},{y[x]},x,IncludeSingularSolutions->True]
\[
y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} -\exp \left (\int _1^x-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right ) \sqrt {3 x^2+1} c_1 \left (\int _1^0\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]-\int _1^x\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]\right ) & 2 \exp \left (\int _1^0-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}+\int _1^1-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right ) \int _1^1\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]-2 \exp \left (\int _1^0-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}+\int _1^1-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right ) \int _1^0\exp \left (-2 \int _1^{K[2]}-\frac {3 i \left (\left (3 \unicode {f80d}^2+1\right )^2 \sqrt {\lambda }-3 i \unicode {f80d}\right )}{9 \unicode {f80d}^2+3}d\unicode {f80d}\right )dK[2]=0 \\ 0 & \text {True} \\ \end {array} \\ \end {array}
\]
✗ Sympy
from sympy import *
x = symbols("x")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(lambda_*(3*x**2 + 1)*y(x) - 6*x*Derivative(y(x), x)/(3*x**2 + 1)**2 + Derivative(y(x), (x, 2))/(3*x**2 + 1),0)
ics = {y(0): 0, y(pi): 0}
dsolve(ode,func=y(x),ics=ics)
ValueError : Couldnt solve for initial conditions