23.5.187 problem 187

Internal problem ID [6796]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 187
Date solved : Tuesday, September 30, 2025 at 03:51:46 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} a y+y^{\left (6\right )}&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 102

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

\[ y = c_1 \,{\mathrm e}^{-\frac {\left (i \sqrt {3}-1\right ) \left (-a \right )^{{1}/{6}} x}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (1+i \sqrt {3}\right ) \left (-a \right )^{{1}/{6}} x}{2}}+c_3 \,{\mathrm e}^{\frac {\left (i \sqrt {3}-1\right ) \left (-a \right )^{{1}/{6}} x}{2}}+c_4 \,{\mathrm e}^{\frac {\left (1+i \sqrt {3}\right ) \left (-a \right )^{{1}/{6}} x}{2}}+c_5 \,{\mathrm e}^{-\left (-a \right )^{{1}/{6}} x}+c_6 \,{\mathrm e}^{\left (-a \right )^{{1}/{6}} x} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 98

ode=a*y[x] + D[y[x],{x,6}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2 e^{(-1)^{5/6} \sqrt [6]{a} x}+c_3 e^{-\sqrt [6]{-1} \sqrt [6]{a} x}+c_5 e^{-(-1)^{5/6} \sqrt [6]{a} x}+c_6 e^{\sqrt [6]{-1} \sqrt [6]{a} x}+c_1 \cos \left (\sqrt [6]{a} x\right )+c_4 \sin \left (\sqrt [6]{a} x\right ) \end{align*}

✓ Sympy. Time used: 0.201 (sec). Leaf size: 114

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*y(x) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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