64.23.6 problem 8

Internal problem ID [13612]
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 : 8
Date solved : Monday, March 31, 2025 at 08:02:42 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^{\prime }\left ({\mathrm e}^{\pi }\right )&=0 \end{align*}

✓ Maple. Time used: 0.033 (sec). Leaf size: 5

ode:=diff(y(x),x)+x*diff(diff(y(x),x),x)+lambda/x*y(x) = 0; 
ic:=y(1) = 0, D(y)(exp(Pi)) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = 0 \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 92

ode=D[x*D[y[x],x],x]+\[Lambda]/x*y[x]==0; 
ic={y[1]==0,Derivative[1][y][Exp[Pi]]==0}; 
DSolve[{ode,ic},{y[x]},x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.439 (sec). Leaf size: 1163

from sympy import * 
x = symbols("x") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(lambda_*y(x)/x + x*Derivative(y(x), (x, 2)) + Derivative(y(x), x),0) 
ics = {y(1): 0, Subs(Derivative(y(x), x), x, exp(pi)): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \text {Solution too large to show} \]