23.3.542 problem 548

Internal problem ID [6256]
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 : 548
Date solved : Tuesday, September 30, 2025 at 02:39:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 52

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

\[ y = \frac {c_2 \left (x^{2}-1\right ) \ln \left (x +\sqrt {x^{2}-1}\right )+c_1 \,x^{2}-c_2 \sqrt {x^{2}-1}\, x -c_1}{\sqrt {x^{2}-1}\, x} \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 77

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

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

✓ Sympy. Time used: 0.251 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \frac {C_{1} \left (x^{2}\right )^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} 1, 1 \\ \frac {5}{2} \end {matrix}\middle | {x^{2}} \right )} + C_{2} {{}_{1}F_{0}\left (\begin {matrix} - \frac {1}{2} \\ \end {matrix}\middle | {x^{2}} \right )}}{\sqrt {x} \sqrt [4]{x^{2}}} \]