2.2.3 Problem 3

Internal problem ID [9126]
Book : Second order enumerated odes
Section : section 2
Problem number : 3
Date solved : Sunday, February 23, 2025 at 05:33:25 AM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

Solve

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

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
<- 2nd_order Liouville successful`
 

Maple dsolve solution

Solving time : 0.010 (sec)
Leaf size : 61

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

\[ c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x -1\right )}{2}\right )-c_{2} +\frac {2 \,3^{{5}/{6}} y \pi }{9 \Gamma \left (\frac {2}{3}\right ) \left (-y^{3}\right )^{{1}/{3}}}-\frac {y \Gamma \left (\frac {1}{3}, -\frac {y^{3}}{3}\right ) 3^{{1}/{3}}}{3 \left (-y^{3}\right )^{{1}/{3}}} = 0 \]

✓Mathematica DSolve solution

Solving time : 16.947 (sec)
Leaf size : 64

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

\[ y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1} \Gamma \left (\frac {1}{3},-\frac {\text {$\#$1}^3}{3}\right )}{3^{2/3} \sqrt [3]{-\text {$\#$1}^3}}\&\right ]\left [\int _1^x-e^{\frac {1}{2} (K[2]-2) K[2]} c_1dK[2]+c_2\right ] \]

✗Sympy solution

Solving time : 0.000 (sec)
Leaf size : 0

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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

NotImplementedError : The given ODE -(x + sqrt(x**2 - 2*x - 4*y(x)**2*Derivative(y(x), (x, 2)) + 1) - 1)/(2*y(x)**2) + Derivative(y(x), x) cannot be solved by the factorable group method