87.22.12 problem 12

Internal problem ID [23757]
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 : 12
Date solved : Thursday, October 02, 2025 at 09:44:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-9 y&=20 \cos \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=18 \\ \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 21

ode:=diff(diff(y(t),t),t)-9*y(t) = 20*cos(t); 
ic:=[y(0) = 0, D(y)(0) = 18]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 24

ode=D[y[t],{t,2}]-9*y[t]==20*Cos[t]; 
ic={y[0]==0,Derivative[1][y][0] ==18}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.074 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-9*y(t) - 20*cos(t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(t), t), t, 0): 18} 
dsolve(ode,func=y(t),ics=ics)
 

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