80.3.25 problem 26

Internal problem ID [21189]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 3. First order nonlinear differential equations. Excercise 3.7 at page 67
Problem number : 26
Date solved : Thursday, October 02, 2025 at 07:16:17 PM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.143 (sec). Leaf size: 77

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

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

✓ Mathematica. Time used: 0.538 (sec). Leaf size: 87

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

\begin{align*} y(x)&\to \sqrt [3]{2-2 x^3}+x\\ y(x)&\to x+\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{1-x^3}}{2^{2/3}}\\ y(x)&\to x+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{1-x^3}}{2^{2/3}} \end{align*}

✗ Sympy

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

NotImplementedError : Initial conditions produced too many solutions for constants