58.23.8 problem 10

Internal problem ID [14966]
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 : Thursday, October 02, 2025 at 09:57:09 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,op(ic)],y(x), singsol=all);
 

\[ y = 0 \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 79

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]
 

\begin{align*} y(x)&\to \begin {array}{cc} \{ & \begin {array}{cc} c_1 \sin \left (x \left (x^2+1\right ) \sqrt {\lambda }\right ) & \unicode {f80d}\in \mathbb {Z}\land \left (\left (\sqrt {\unicode {f80d}^2}=\unicode {f80d}\land \unicode {f80d}^2 \pi ^2=\lambda \right )\lor \left (\sqrt {(2 \unicode {f80d}+1)^2}=2 \unicode {f80d}+1\land (2 \pi \unicode {f80d}+\pi )^2=4 \lambda \right )\right ) \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ 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