25.6.18 problem 18

Internal problem ID [6902]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 04:03:15 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} y y^{\prime }&=x +y^{2}-y^{2} {y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.167 (sec). Leaf size: 77

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

\begin{align*} y &= -\frac {\sqrt {-4 x -1}}{2} \\ y &= \frac {\sqrt {-4 x -1}}{2} \\ y &= -\frac {\sqrt {4 x^{2}+\left (-8 c_1 -4\right ) x +4 c_1^{2}-1}}{2} \\ y &= \frac {\sqrt {4 x^{2}+\left (-8 c_1 -4\right ) x +4 c_1^{2}-1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to -\frac {1}{2} \sqrt {4 x^2-4 (1+4 c_1) x-1+16 c_1{}^2}\\ y(x)&\to \frac {1}{2} \sqrt {4 x^2-4 (1+4 c_1) x-1+16 c_1{}^2} \end{align*}

✓ Sympy. Time used: 3.899 (sec). Leaf size: 78

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

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {- 4 x + 4 \left (C_{1} + x\right )^{2} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- 4 x + 4 \left (C_{1} + x\right )^{2} - 1}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {- 4 x + 4 \left (C_{1} + x\right )^{2} - 1}}{2}, \ y{\left (x \right )} = \frac {\sqrt {- 4 x + 4 \left (C_{1} + x\right )^{2} - 1}}{2}\right ] \]