67.20.4 problem 29.6 (d)

Internal problem ID [16900]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 29. Convolution. Additional Exercises. page 523
Problem number : 29.6 (d)
Date solved : Thursday, October 02, 2025 at 01:40:13 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=\sin \left (2 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: 0.098 (sec). Leaf size: 18

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

\[ y = \frac {\sin \left (2 t \right )}{8}-\frac {\cos \left (2 t \right ) t}{4} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 93

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

\begin{align*} y(t)&\to \sin (2 t) \left (\int _1^t\frac {1}{4} \sin (4 K[2])dK[2]-\int _1^0\frac {1}{4} \sin (4 K[2])dK[2]\right )-\cos (2 t) \int _1^0-\frac {1}{2} \sin ^2(2 K[1])dK[1]+\cos (2 t) \int _1^t-\frac {1}{2} \sin ^2(2 K[1])dK[1] \end{align*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*y(t) - sin(2*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)
 

\[ y{\left (t \right )} = - \frac {t \cos {\left (2 t \right )}}{4} + \frac {\sin {\left (2 t \right )}}{8} \]