15.19.10 problem 10

Internal problem ID [3294]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 10
Date solved : Sunday, March 30, 2025 at 01:32:56 AM
CAS classification : [_quadrature]

\begin{align*} y {y^{\prime }}^{2}+\left (y^{2}-x^{3}-x y^{2}\right ) y^{\prime }-x y \left (x^{2}+y^{2}\right )&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 48

ode:=y(x)*diff(y(x),x)^2+(y(x)^2-x^3-x*y(x)^2)*diff(y(x),x)-x*y(x)*(x^2+y(x)^2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= c_1 \,{\mathrm e}^{-x} \\ y &= \sqrt {{\mathrm e}^{x^{2}} c_1 -x^{2}-1} \\ y &= -\sqrt {{\mathrm e}^{x^{2}} c_1 -x^{2}-1} \\ \end{align*}

✓ Mathematica. Time used: 15.082 (sec). Leaf size: 61

ode=y[x]*D[y[x],x]^2+(y[x]^2-x^3-x*y[x]^2)*D[y[x],x]-x*y[x]*(x^2+y[x]^2)==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.987 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x**2 + y(x)**2)*y(x) + (-x**3 - x*y(x)**2 + y(x)**2)*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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