75.22.12 problem 717

Internal problem ID [17112]
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 : 717
Date solved : Thursday, March 13, 2025 at 09:16:25 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\lambda ^{2} y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 5

ode:=diff(diff(y(x),x),x)+lambda^2*y(x) = 0; 
ic:=D(y)(0) = 0, D(y)(Pi) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = 0 \]

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

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

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

✓ Sympy. Time used: 0.111 (sec). Leaf size: 3

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

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