14.33.4 problem 4

Internal problem ID [2838]
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 : 4
Date solved : Monday, January 27, 2025 at 06:19:21 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 \left (L \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 5

dsolve([diff(y(t),t$2)+lambda*y(t)=0,D(y)(0) = 0, y(L) = 0],y(t), singsol=all)
 

\[ y = 0 \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 46

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} c_1 \cos \left (t \sqrt {\lambda }\right ) & \unicode {f80d}\in \mathbb {Z}\land \unicode {f80d}\geq 1\land \lambda =\frac {\left (\unicode {f80d}-\frac {1}{2}\right )^2 \pi ^2}{L^2}\land L>0 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]