85.70.3 problem 5 (c)

Internal problem ID [22927]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 6. Solution of linear differential equations by Laplace transform. A Exercises at page 283
Problem number : 5 (c)
Date solved : Thursday, October 02, 2025 at 09:16:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+9 y&=20 \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)+9*y(t) = 20*exp(-t); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 23

ode=D[y[t],{t,2}]+9*y[t]==20*Exp[-t]; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.066 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(9*y(t) + Derivative(y(t), (t, 2)) - 20*exp(-t),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

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