89.13.8 problem 8

Internal problem ID [24591]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 8. Linear Differential Equations with constant coefficients. Exercises at page 127
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:46:20 PM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }+2 y^{\prime \prime }+y^{\prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

ode:=diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x)+diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{-2 x}+c_2 \sin \left (x \right )+c_3 \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 24

ode=D[y[x],{x,3}]+2*D[y[x],{x,2}]+D[y[x],x]+2*y[x] ==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_3 e^{-2 x}+c_1 \cos (x)+c_2 \sin (x) \end{align*}

✓ Sympy. Time used: 0.072 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x) + Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- 2 x} + C_{2} \sin {\left (x \right )} + C_{3} \cos {\left (x \right )} \]