58.23.2 problem 2

Internal problem ID [14960]
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 : 2
Date solved : Thursday, October 02, 2025 at 09:57:02 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

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

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

\[ y = 0 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 36

ode=D[y[x],{x,2}]+\[Lambda]*y[x]==0; 
ic={y[0]==0,Derivative[1][y][Pi]==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 \sqrt {\lambda }\right ) & \unicode {f80d}\in \mathbb {Z}\land \unicode {f80d}\geq 1\land \lambda =\left (\unicode {f80d}-\frac {1}{2}\right )^2 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 3

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

\[ y{\left (x \right )} = 0 \]