23.1.558 problem 548

Internal problem ID [5165]
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 : 548
Date solved : Tuesday, September 30, 2025 at 11:47:19 AM
CAS classification : [_rational, _Bernoulli]

\begin{align*} 2 x y y^{\prime }+x^{2} \left (a \,x^{3}+1\right )&=6 y^{2} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

ode:=2*x*y(x)*diff(y(x),x)+x^2*(a*x^3+1) = 6*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {\sqrt {4 c_1 \,x^{4}+4 a \,x^{3}+1}\, x}{2} \\ y &= \frac {\sqrt {4 c_1 \,x^{4}+4 a \,x^{3}+1}\, x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.67 (sec). Leaf size: 59

ode=2*x*y[x]*D[y[x],x]+x^2*(1+a*x^3)==6*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} \sqrt {4 a x^5+4 c_1 x^6+x^2}\\ y(x)&\to \frac {1}{2} \sqrt {4 a x^5+4 c_1 x^6+x^2} \end{align*}

✓ Sympy. Time used: 0.351 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*(a*x**3 + 1) + 2*x*y(x)*Derivative(y(x), x) - 6*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {x \sqrt {C_{1} x^{4} + 4 a x^{3} + 1}}{2}, \ y{\left (x \right )} = \frac {x \sqrt {C_{1} x^{4} + 4 a x^{3} + 1}}{2}\right ] \]