84.38.12 problem 26.24

Internal problem ID [22366]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Supplementary problems
Problem number : 26.24
Date solved : Thursday, October 02, 2025 at 08:37:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+5 y&=3 \,{\mathrm e}^{-2 t} \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: 0.048 (sec). Leaf size: 30

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+5*y(t) = 3*exp(-2*t); 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = \frac {3 \,{\mathrm e}^{-2 t}}{5}+\frac {{\mathrm e}^{-t} \left (4 \cos \left (2 t \right )+13 \sin \left (2 t \right )\right )}{10} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 34

ode=D[y[t],{t,2}]+2*D[y[t],{t,1}]+5*y[t]==3*Exp[-2*t]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{10} e^{-2 t} \left (13 e^t \sin (2 t)+4 e^t \cos (2 t)+6\right ) \end{align*}

✓ Sympy. Time used: 0.170 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) + 2*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 3*exp(-2*t),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (\frac {13 \sin {\left (2 t \right )}}{10} + \frac {2 \cos {\left (2 t \right )}}{5} + \frac {3 e^{- t}}{5}\right ) e^{- t} \]