67.4.10 problem Problem 2(i)[j]

Internal problem ID [13983]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(i)[j]
Date solved : Monday, March 31, 2025 at 08:20:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+8 y^{\prime }+20 y&=\sin \left (2 t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-4 \end{align*}

✓ Maple. Time used: 0.102 (sec). Leaf size: 31

ode:=diff(diff(y(t),t),t)+8*diff(y(t),t)+20*y(t) = sin(2*t); 
ic:=y(0) = 1, D(y)(0) = -4; 
dsolve([ode,ic],y(t),method='laplace');
 

\[ y = \frac {\cos \left (2 t \right ) \left (-1+33 \,{\mathrm e}^{-4 t}\right )}{32}+\frac {\sin \left (2 t \right ) \left (1+{\mathrm e}^{-4 t}\right )}{32} \]

✓ Mathematica. Time used: 0.275 (sec). Leaf size: 124

ode=D[y[t],{t,2}]+8*D[y[t],t]+20*y[t]==Sin[2*t]; 
ic={y[0]==1,Derivative[1][y][0] ==-4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.280 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(20*y(t) - sin(2*t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): -4} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {\sin {\left (2 t \right )}}{32} + \frac {33 \cos {\left (2 t \right )}}{32}\right ) e^{- 4 t} + \frac {\sin {\left (2 t \right )}}{32} - \frac {\cos {\left (2 t \right )}}{32} \]