90.11.3 problem 37

Internal problem ID [25201]
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 : 37
Date solved : Thursday, October 02, 2025 at 11:57:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y&=4 t^{2} \cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 14

ode:=diff(diff(y(t),t),t)-3*y(t) = 4*t^2*cos(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = t \left (-t \cos \left (t \right )+\sin \left (t \right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 15

ode=D[y[t],{t,2}]-3*y[t]==4*t^2*Cos[t]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.091 (sec). Leaf size: 14

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-4*t**2*cos(t) - 3*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

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