90.11.1 problem 35

Internal problem ID [25199]
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 : 35
Date solved : Thursday, October 02, 2025 at 11:57:44 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 15

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

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

✓ Sympy. Time used: 0.057 (sec). Leaf size: 14

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

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