90.26.8 problem 30

Internal problem ID [25409]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 6. Discontinuous Functions and the Laplace Transform. Exercises at page 379
Problem number : 30
Date solved : Friday, October 03, 2025 at 12:01:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=\left \{\begin {array}{cc} 0 & 0\le t <2 \\ 4 & 2\le t <\infty \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✗ Maple

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+4*y(t) = piecewise(0 <= t and t < 2,0,2 <= t and t < infinity,4); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 48

ode=D[y[t],{t,2}]-4*D[y[t],t]+4*y[t]==Piecewise[{ {0,0<=t<2}, {4,2<=t<Infinity}}]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} e^{2 t} (1-2 t) & t\leq 2 \\ e^{2 t} (1-2 t)+e^{2 t-4} (2 t-5)+1 & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((0, (t >= 0) & (t < 2)), (4, (t >= 2) & (t < oo))) + 4*y(t) - 4*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)