90.11.4 problem 38

Internal problem ID [25202]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 2. The Laplace Transform. Exercises at page 163
Problem number : 38
Date solved : Thursday, October 02, 2025 at 11:57:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=32 t \cos \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.105 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+4*y(t) = 32*t*cos(2*t); 
ic:=[y(0) = 0, D(y)(0) = 2]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = 2 t \cos \left (2 t \right )+4 \sin \left (2 t \right ) t^{2} \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 20

ode=D[y[t],{t,2}]+4*y[t]==32*t*Cos[2*t]; 
ic={y[0]==0,Derivative[1][y][0] ==2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 t (2 t \sin (2 t)+\cos (2 t)) \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-32*t*cos(2*t) + 4*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = 4 t^{2} \sin {\left (2 t \right )} + 2 t \cos {\left (2 t \right )} \]