8.10.5 problem 5

Internal problem ID [2587]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.4. The method of variation of parameters. Excercises page 156
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 05:47:29 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 3 y^{\prime \prime }+4 y^{\prime }+y&=\sin \left (t \right ) {\mathrm e}^{-t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 26

ode:=3*diff(diff(y(t),t),t)+4*diff(y(t),t)+y(t) = sin(t)*exp(-t); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\left (24 \,{\mathrm e}^{\frac {2 t}{3}}+2 \cos \left (t \right )-3 \sin \left (t \right )-13\right ) {\mathrm e}^{-t}}{13} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 33

ode=3*D[y[t],{t,2}]+4*D[y[t],t]+y[t]==Sin[t]*Exp[-t]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{13} e^{-t} \left (24 e^{2 t/3}-3 \sin (t)+2 \cos (t)-13\right ) \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) + 4*Derivative(y(t), t) + 3*Derivative(y(t), (t, 2)) - exp(-t)*sin(t),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (- \frac {3 \sin {\left (t \right )}}{13} + \frac {2 \cos {\left (t \right )}}{13} - 1\right ) e^{- t} + \frac {24 e^{- \frac {t}{3}}}{13} \]