64.23.5 problem 7

Internal problem ID [13682]
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 : 7
Date solved : Tuesday, January 28, 2025 at 05:55:06 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime }+x y^{\prime \prime }+\frac {\lambda y}{x}&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0\\ y \left ({\mathrm e}^{\pi }\right )&=0 \end{align*}

Solution by Maple

Time used: 0.011 (sec). Leaf size: 5

dsolve([diff(x*diff(y(x),x),x)+lambda/x*y(x)=0,y(1) = 0, y(exp(Pi)) = 0],y(x), singsol=all)
 
\[ y = 0 \]

Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 70

DSolve[{D[x*D[y[x],x],x]+\[Lambda]/x*y[x]==0,{y[1]==0,y[Exp[Pi]]==0}},{y[x]},x,IncludeSingularSolutions -> True]
 
\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} c_1 \sin \left (\sqrt {\lambda } \log (x)\right ) & \unicode {f80d}\in \mathbb {Z}\land \left (\left (\sqrt {\unicode {f80d}^2}=\unicode {f80d}\land 4 \unicode {f80d}^2=\lambda \right )\lor \left (\sqrt {(2 \unicode {f80d}+1)^2}=2 \unicode {f80d}+1\land (2 \unicode {f80d}+1)^2=\lambda \right )\right ) \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]