69.22.13 problem 718

Internal problem ID [18479]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 17. Boundary value problems. Exercises page 163
Problem number : 718
Date solved : Thursday, October 02, 2025 at 03:14:08 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\lambda ^{2} 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: 0.036 (sec). Leaf size: 5

ode:=diff(diff(y(x),x),x)+lambda^2*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: 40

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

✓ Sympy. Time used: 0.063 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq(lambda_**2*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)
 

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