61.4.12 problem Problem 2(k)[l]

Internal problem ID [15337]
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(k)[l]
Date solved : Thursday, October 02, 2025 at 10:11:45 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.098 (sec). Leaf size: 25

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

\[ y = -\frac {2 \cos \left (t \right )}{5}-\frac {6 \sin \left (t \right )}{5}+2 \,{\mathrm e}^{-t}-\frac {8 \,{\mathrm e}^{\frac {t}{2}}}{5} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 34

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

\begin{align*} y(t)&\to \frac {2}{5} \left (5 e^{-t}-4 e^{t/2}-3 \sin (t)-\cos (t)\right ) \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = - \frac {8 e^{\frac {t}{2}}}{5} - \frac {6 \sin {\left (t \right )}}{5} - \frac {2 \cos {\left (t \right )}}{5} + 2 e^{- t} \]