2.2.3 Problem 3

Internal problem ID [9126]
Book : Second order enumerated odes
Section : section 2
Problem number : 3
Date solved : Sunday, March 30, 2025 at 02:17:05 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

✓ Maple. Time used: 0.021 (sec). Leaf size: 61

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

$$c_1\mathrm{erf}\left(\frac{i\sqrt{2}(x-1)}{2}\right)-c_2+\frac{23^{5/6}y\pi }{9\mathrm{\Gamma }\left(\frac{2}{3}\right){(-y^3)}^{1/3}}-\frac{y\mathrm{\Gamma }(\frac{1}{3},-\frac{y^3}{3})3^{1/3}}{3{(-y^3)}^{1/3}}=0$$

Maple trace

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

✓ Mathematica. Time used: 16.947 (sec). Leaf size: 64

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

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

✗ Sympy

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