64.23.4 problem 4

Internal problem ID [13610]
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 : 4
Date solved : Monday, March 31, 2025 at 08:02:37 AM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 5

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

\[ y = 0 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

ode=D[y[x],{x,2}]+\[Lambda]*y[x]==0; 
ic={Derivative[1][y][0]==0,Derivative[1][y][L]==0}; 
DSolve[{ode,ic},{y[x]},x,IncludeSingularSolutions->True]
 

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} c_1 \cos \left (x \sqrt {\lambda }\right ) & \unicode {f80d}\in \mathbb {Z}\land \unicode {f80d}\geq 0\land \lambda =\frac {\unicode {f80d}^2 \pi ^2}{L^2}\land L>0 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.106 (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 = {Subs(Derivative(y(x), x), x, 0): 0, Derivative(y(L), L): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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