63.1.21 problem 24

Internal problem ID [15461]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 24
Date solved : Thursday, October 02, 2025 at 10:15:18 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 50

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

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

✓ Mathematica. Time used: 0.242 (sec). Leaf size: 74

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

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

✓ Sympy. Time used: 0.558 (sec). Leaf size: 32

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

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