23.6.21 problem 21

Internal problem ID [6820]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 6. THE EQUATION IS NONLINEAR AND OF ORDER GREATER THAN TWO, page 427
Problem number : 21
Date solved : Tuesday, September 30, 2025 at 03:52:35 PM
CAS classification : [[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 94

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

\begin{align*} y &= \frac {\left (x^{2}+2\right ) \sqrt {-x^{2}+1}}{6}+c_{1} x +\frac {x \arcsin \left (x \right )}{2}+c_{2} \\ y &= -\frac {\sqrt {-x^{2}+1}\, x^{2}}{6}-\frac {\sqrt {-x^{2}+1}}{3}-\frac {x \arcsin \left (x \right )}{2}+c_{1} x +c_{2} \\ y &= \frac {\sqrt {c_{1}^{2}-1}\, x^{3}}{6}+\frac {c_{1} x^{2}}{2}+c_{2} x +c_{3} \\ \end{align*}

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 75

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

\begin{align*} y(x)&\to \frac {c_1 x^3}{6}-\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2\\ y(x)&\to \frac {c_1 x^3}{6}+\frac {1}{2} \sqrt {1+c_1{}^2} x^2+c_3 x+c_2 \end{align*}

✗ Sympy

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

Timed Out