58.23.7 problem 9

Internal problem ID [14965]
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 : Thursday, October 02, 2025 at 09:57:08 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,op(ic)],y(x), singsol=all);
 

\[ y = 0 \]

✓ Mathematica. Time used: 1.025 (sec). Leaf size: 72

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]
 

\begin{align*} y(x)&\to \begin {array}{cc} \{ & \begin {array}{cc} c_1 \sin \left (\sqrt {\lambda } \arctan (x)\right ) & \unicode {f80d}\in \mathbb {Z}\land \left (\left (\sqrt {\unicode {f80d}^2}=\unicode {f80d}\land 64 \unicode {f80d}^2=\lambda \right )\lor \left (\sqrt {(2 \unicode {f80d}+1)^2}=2 \unicode {f80d}+1\land 16 (2 \unicode {f80d}+1)^2=\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_*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