23.5.8 problem 8

Internal problem ID [6617]
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 : 8
Date solved : Tuesday, September 30, 2025 at 03:50:17 PM
CAS classification : [[_3rd_order, _missing_x]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to c_1 e^{-(-1)^{2/3} \sqrt [3]{a} x}+c_2 e^{\sqrt [3]{-1} \sqrt [3]{a} x}+c_3 e^{-\sqrt [3]{a} x} \end{align*}

✓ Sympy. Time used: 0.077 (sec). Leaf size: 58

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

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