89.14.25 problem 25

Internal problem ID [24629]
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 128
Problem number : 25
Date solved : Thursday, October 02, 2025 at 10:46:35 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+2 y^{\prime \prime }-8 y^{\prime }-8 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+4*diff(diff(diff(y(x),x),x),x)+2*diff(diff(y(x),x),x)-8*diff(y(x),x)-8*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_3 \,{\mathrm e}^{\sqrt {2}\, x}+c_4 \,{\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{-2 x} \left (c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 44

ode=D[y[x],{x,4}]+4*D[y[x],{x,3}]+2*D[y[x],{x,2}]-8*D[y[x],{x,1}]-8*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 e^{\sqrt {2} x}+c_2 e^{-\sqrt {2} x}+e^{-2 x} (c_4 x+c_3) \end{align*}

✓ Sympy. Time used: 0.158 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x) - 8*Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} e^{- \sqrt {2} x} + C_{4} e^{\sqrt {2} x} + \left (C_{1} + C_{2} x\right ) e^{- 2 x} \]