23.1.690 problem 685

Internal problem ID [5297]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 685
Date solved : Sunday, October 12, 2025 at 01:25:06 AM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 122

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

\begin{align*} y &= \frac {\sqrt {-\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) c_1 \,{\mathrm e}^{2}\right ) \left (x^{2} \operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) c_1 \,{\mathrm e}^{2}\right )-2 x^{2}+a \right )}}{\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) c_1 \,{\mathrm e}^{2}\right )} \\ y &= -\frac {\sqrt {-\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) c_1 \,{\mathrm e}^{2}\right ) \left (x^{2} \operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) c_1 \,{\mathrm e}^{2}\right )-2 x^{2}+a \right )}}{\operatorname {LambertW}\left (-\left (-2 x^{2}+a \right ) c_1 \,{\mathrm e}^{2}\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.188 (sec). Leaf size: 148

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {K[2]}{x^2+K[2]^2}+\frac {\left (2 x^2-a\right ) K[2]}{\left (x^2+K[2]^2\right )^2}-\int _1^x\left (\frac {4 K[1] K[2] \left (2 K[2]^2+a\right )}{\left (K[1]^2+K[2]^2\right )^3}-\frac {6 K[1] K[2]}{\left (K[1]^2+K[2]^2\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\left (\frac {K[1]}{K[1]^2+y(x)^2}-\frac {K[1] \left (2 y(x)^2+a\right )}{\left (K[1]^2+y(x)^2\right )^2}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out