86.7.9 problem 11

Internal problem ID [23156]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5c at page 83
Problem number : 11
Date solved : Thursday, October 02, 2025 at 09:23:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+9 x^{\prime }+8 x&=\sin \left (5 t \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

ode:=diff(diff(x(t),t),t)+9*diff(x(t),t)+8*x(t) = sin(5*t); 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{-t} c_2 +{\mathrm e}^{-8 t} c_1 -\frac {45 \cos \left (5 t \right )}{2314}-\frac {17 \sin \left (5 t \right )}{2314} \]

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 39

ode=D[x[t],{t,2}]+9*D[x[t],t]+8*x[t]==Sin[5*t]; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 e^{-8 t}+c_2 e^{-t}+\frac {-17 \sin (5 t)-45 \cos (5 t)}{2314} \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(8*x(t) - sin(5*t) + 9*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = C_{1} e^{- 8 t} + C_{2} e^{- t} - \frac {17 \sin {\left (5 t \right )}}{2314} - \frac {45 \cos {\left (5 t \right )}}{2314} \]