14.33.2 problem 2

Internal problem ID [2836]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 5. Separation of variables and Fourier series. Section 5.1 (Two point boundary-value problems). Page 480
Problem number : 2
Date solved : Sunday, March 30, 2025 at 12:33:35 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: 0.030 (sec). Leaf size: 5

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

\[ y = 0 \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

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

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

✓ Sympy. Time used: 0.126 (sec). Leaf size: 3

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

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