90.26.1 problem 17 (a)

Internal problem ID [25402]
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 : 17 (a)
Date solved : Friday, October 03, 2025 at 12:01:09 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\left \{\begin {array}{cc} 4 & 0\le t <2 \\ 8 t & 2\le t <\infty \end {array}\right . \end{align*}

Using Laplace method

✗ Maple

ode:=diff(diff(y(t),t),t)+4*y(t) = piecewise(0 <= t and t < 2,4,2 <= t and t < infinity,8*t); 
dsolve(ode,y(t),method='laplace');
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.144 (sec). Leaf size: 85

ode=D[y[t],{t,2}]+4*y[t]==Piecewise[{{4,0<=t<2},{8*t,2<=t<Infinity}}]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \begin {array}{cc} \{ & \begin {array}{cc} c_1 \cos (2 t)+c_2 \sin (2 t) & t\leq 0 \\ (c_1-1) \cos (2 t)+c_2 \sin (2 t)+1 & 0<t\leq 2 \\ 2 t-3 \cos (4-2 t)+(c_1-1) \cos (2 t)+\sin (4-2 t)+c_2 \sin (2 t) & \text {True} \\ \end {array} \\ \end {array} \end{align*}

✓ Sympy. Time used: 0.353 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-Piecewise((4, (t >= 0) & (t < 2)), (8*t, (t >= 2) & (t < oo))) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} \sin {\left (2 t \right )} + C_{2} \cos {\left (2 t \right )} + \begin {cases} 1 & \text {for}\: t \geq 0 \wedge t < 2 \\2 t & \text {for}\: t \geq 2 \wedge t < \infty \\\text {NaN} & \text {otherwise} \end {cases} \]