23.3.541 problem 547

Internal problem ID [6255]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 547
Date solved : Tuesday, September 30, 2025 at 02:39:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -x^{2} y-\left (-x^{3}+1\right ) y^{\prime }+x \left (x^{3}+1\right ) y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 54

ode:=-x^2*y(x)-(-x^3+1)*diff(y(x),x)+x*(x^3+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 c_1 \,x^{2} \operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {2}{3}, \frac {-x^{3}+1}{x^{3}+1}\right ) \Gamma \left (\frac {2}{3}\right )}{3 \left (x^{3}+1\right )^{{1}/{3}} \left (-x^{3}\right )^{{1}/{3}}}+c_2 \left (x^{3}+1\right )^{{1}/{3}} \]

✓ Mathematica. Time used: 5.4 (sec). Leaf size: 44

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

\begin{align*} y(x)&\to \frac {1}{2} \sqrt [3]{x^3+1} \left (c_2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},-x^3\right )+2 c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*y(x) + x*(x**3 + 1)*Derivative(y(x), (x, 2)) - (1 - x**3)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False