78.15.15 problem 15

Internal problem ID [18300]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 22. Higher Order Linear Equations. Coupled Harmonic Oscillators. Problems at page 160
Problem number : 15
Date solved : Monday, March 31, 2025 at 05:25:07 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (5\right )}-6 y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+48 y^{\prime \prime }+16 y^{\prime }-96 y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 30

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)-6*diff(diff(diff(diff(y(x),x),x),x),x)-8*diff(diff(diff(y(x),x),x),x)+48*diff(diff(y(x),x),x)+16*diff(y(x),x)-96*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-2 x} \left (c_1 \,{\mathrm e}^{8 x}+\left (c_3 x +c_2 \right ) {\mathrm e}^{4 x}+c_5 x +c_4 \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 43

ode=D[y[x],{x,5}]-6*D[y[x],{x,4}]-8*D[y[x],{x,3}]+48*D[y[x],{x,2}]+16*D[y[x],x]-96*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-2 x} \left (c_2 x+c_3 e^{4 x}+c_4 e^{4 x} x+c_5 e^{8 x}+c_1\right ) \]

✓ Sympy. Time used: 0.262 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-96*y(x) + 16*Derivative(y(x), x) + 48*Derivative(y(x), (x, 2)) - 8*Derivative(y(x), (x, 3)) - 6*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{5} e^{6 x} + \left (C_{1} + C_{2} x\right ) e^{- 2 x} + \left (C_{3} + C_{4} x\right ) e^{2 x} \]