86.7.8 problem 8

Internal problem ID [23155]
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 : 8
Date solved : Thursday, October 02, 2025 at 09:23:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+5 y^{\prime }+8 y&=4 \sin \left (5 x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 43

ode:=diff(diff(y(x),x),x)+5*diff(y(x),x)+8*y(x) = 4*sin(5*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {5 x}{2}} \sin \left (\frac {\sqrt {7}\, x}{2}\right ) c_2 +{\mathrm e}^{-\frac {5 x}{2}} \cos \left (\frac {\sqrt {7}\, x}{2}\right ) c_1 -\frac {50 \cos \left (5 x \right )}{457}-\frac {34 \sin \left (5 x \right )}{457} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 65

ode=D[y[x],{x,2}]+5*D[y[x],x]+8*y[x]==4*Sin[5*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {2}{457} (17 \sin (5 x)+25 \cos (5 x))+c_2 e^{-5 x/2} \cos \left (\frac {\sqrt {7} x}{2}\right )+c_1 e^{-5 x/2} \sin \left (\frac {\sqrt {7} x}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.159 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*y(x) - 4*sin(5*x) + 5*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {7} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {7} x}{2} \right )}\right ) e^{- \frac {5 x}{2}} - \frac {34 \sin {\left (5 x \right )}}{457} - \frac {50 \cos {\left (5 x \right )}}{457} \]