23.3.349 problem 352

Internal problem ID [6063]
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 : 352
Date solved : Tuesday, September 30, 2025 at 02:20:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 49

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

\[ y = \frac {\sqrt {-x^{2}+1}\, \left (x^{3}-x \right )}{9}+\left (\ln \left (x +\sqrt {x^{2}-1}\right ) c_1 +c_2 \right ) x -\sqrt {x^{2}-1}\, c_1 \]

✓ Mathematica. Time used: 0.217 (sec). Leaf size: 173

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

\begin{align*} y(x)&\to \frac {\left (x^2-1\right ) \left (x^5-2 x^3-9 c_1 \sqrt {1-x^2} x+9 c_2 \sqrt {-\left (x^2-1\right )^2}+x\right )-3 x \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right ) \left (3 c_2 \sqrt {1-x^2} x^2-3 c_2 \sqrt {1-x^2}+\sqrt {x^2-1} x^5-\left (\sqrt {x^2-1}-\sqrt {1-x^2} \sqrt {-\left (x^2-1\right )^2}\right ) x^3\right )}{9 \left (1-x^2\right )^{3/2}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x*(-x**2*sqrt(1 - x**2) + x*Derivative(y(