23.5.72 problem 72

Internal problem ID [6681]
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 : 72
Date solved : Friday, October 03, 2025 at 02:09:43 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x y+3 y^{\prime }+x y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.171 (sec). Leaf size: 121

ode:=x*y(x)+3*diff(y(x),x)+x*diff(diff(diff(y(x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-x} \left (c_2 \int \frac {x \,{\mathrm e}^{-\frac {x \left (i \sqrt {3}-3\right )}{2}} \operatorname {HeunC}\left (i \sqrt {3}, 1, -2, -\frac {3}{2}, \frac {7}{2}, -x \right )}{\left (x +1\right )^{2}}d x +c_3 \int \frac {x \,{\mathrm e}^{-\frac {x \left (i \sqrt {3}-3\right )}{2}} \operatorname {HeunC}\left (i \sqrt {3}, 1, -2, -\frac {3}{2}, \frac {7}{2}, -x \right ) \int \frac {\left (x +1\right ) {\mathrm e}^{i \sqrt {3}\, x}}{x^{2} \operatorname {HeunC}\left (i \sqrt {3}, 1, -2, -\frac {3}{2}, \frac {7}{2}, -x \right )^{2}}d x}{\left (x +1\right )^{2}}d x +c_1 \right ) \left (x +1\right ) \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -x*(-y(x) - Derivative(y(x), (x, 3)))/3 + Derivative(y(x), x) ca