90.11.2 problem 36

Internal problem ID [25200]
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 : 36
Date solved : Thursday, October 02, 2025 at 11:57:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=36 t \sin \left (3 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 22

ode:=diff(diff(y(t),t),t)+9*y(t) = 36*t*sin(3*t); 
ic:=[y(0) = 0, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -3 \cos \left (3 t \right ) t^{2}+\sin \left (3 t \right ) \left (1+t \right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 23

ode=D[y[t],{t,2}]+9*y[t]==36*t*Sin[3*t]; 
ic={y[0]==0,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to (t+1) \sin (3 t)-3 t^2 \cos (3 t) \end{align*}

✓ Sympy. Time used: 0.083 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-36*t*sin(3*t) + 9*y(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 3} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - 3 t^{2} \cos {\left (3 t \right )} + \left (t + 1\right ) \sin {\left (3 t \right )} \]