87.22.29 problem 29

Internal problem ID [23774]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 4. The Laplace transform. Exercise at page 199
Problem number : 29
Date solved : Thursday, October 02, 2025 at 09:45:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=39 \,{\mathrm e}^{t} \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: 0.072 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t)+5*y(t) = 39*exp(t)*sin(t); 
ic:=[y(0) = -1, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 36

ode=D[y[t],{t,2}]+4*D[y[t],t]+5*y[t]==39*Exp[t]*Sin[t]; 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.210 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 39*exp(t)*sin(t) + 4*Derivative(y(t), 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)
 

\[ y{\left (t \right )} = \left (4 \sin {\left (t \right )} + 3 \cos {\left (t \right )}\right ) e^{- 2 t} + 3 e^{t} \sin {\left (t \right )} - 2 e^{t} \cos {\left (t \right )} \]