14.33.5 problem 8

Internal problem ID [2839]
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 : 8
Date solved : Sunday, March 30, 2025 at 12:33:42 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+\left (1+\lambda \right ) y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y \left (1\right )&=0 \end{align*}

✓ Maple. Time used: 0.066 (sec). Leaf size: 5

ode:=diff(diff(y(t),t),t)-2*diff(y(t),t)+(1+lambda)*y(t) = 0; 
ic:=y(0) = 0, y(1) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = 0 \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 36

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

\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} 2 e^t c_1 \sinh \left (t \sqrt {-\lambda }\right ) & \sinh \left (\sqrt {-\lambda }\right )=0 \\ 0 & \text {True} \\ \end {array} \\ \end {array} \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 3

from sympy import * 
t = symbols("t") 
lambda_ = symbols("lambda_") 
y = Function("y") 
ode = Eq((lambda_ + 1)*y(t) - 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, y(1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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