23.1.641 problem 635

Internal problem ID [5248]
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 : 635
Date solved : Tuesday, September 30, 2025 at 12:00:12 PM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]

\begin{align*} \left (x -6 y\right )^{2} y^{\prime }+a +2 x y-6 y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 113

ode:=(x-6*y(x))^2*diff(y(x),x)+a+2*x*y(x)-6*y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (-x^{3}-18 a x -18 c_1 \right )^{{1}/{3}}}{6}+\frac {x}{6} \\ y &= -\frac {\left (-x^{3}-18 a x -18 c_1 \right )^{{1}/{3}}}{12}-\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 c_1 \right )^{{1}/{3}}}{12}+\frac {x}{6} \\ y &= -\frac {\left (-x^{3}-18 a x -18 c_1 \right )^{{1}/{3}}}{12}+\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 c_1 \right )^{{1}/{3}}}{12}+\frac {x}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.448 (sec). Leaf size: 115

ode=(x-6*y[x])^2*D[y[x],x]+a+2*x*y[x]-6*y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} \left (x+\sqrt [3]{-18 a x-x^3+18 c_1}\right )\\ y(x)&\to \frac {x}{6}+\frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-18 a x-x^3+18 c_1}\\ y(x)&\to \frac {x}{6}-\frac {1}{12} \left (1+i \sqrt {3}\right ) \sqrt [3]{-18 a x-x^3+18 c_1} \end{align*}

✓ Sympy. Time used: 2.100 (sec). Leaf size: 83

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a + 2*x*y(x) + (x - 6*y(x))**2*Derivative(y(x), x) - 6*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x}{6} + \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}}{2}, \ y{\left (x \right )} = \frac {x}{6} + \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}}{2}, \ y{\left (x \right )} = \frac {x}{6} + \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}\right ] \]