23.3.361 problem 365

Internal problem ID [6075]
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 : 365
Date solved : Tuesday, September 30, 2025 at 02:21:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = c_2 x +c_1 \,x^{2}+c_1 -\frac {1}{2}-\frac {1}{6} x^{4} \]

✓ Mathematica. Time used: 0.13 (sec). Leaf size: 109

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

\begin{align*} y(x)&\to \frac {x^6-4 x^4+2 x^3+3 x^2 \left (1+2 c_1 \sqrt {-\left (x^2-1\right )^2}\right )-2 x \left (6 c_1 \sqrt {-\left (x^2-1\right )^2}-3 c_2 \sqrt {-\left (x^2-1\right )^2}+1\right )+6 c_1 \sqrt {-\left (x^2-1\right )^2}}{6-6 x^2} \end{align*}

✗ Sympy

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

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