23.2.240 problem 246

Internal problem ID [5595]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 246
Date solved : Tuesday, September 30, 2025 at 01:10:47 PM
CAS classification : [_separable]

\begin{align*} 4 y^{2} {y^{\prime }}^{2}+2 \left (1+3 x \right ) x y y^{\prime }+3 x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 59

ode:=4*y(x)^2*diff(y(x),x)^2+2*(3*x+1)*x*y(x)*diff(y(x),x)+3*x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 81

ode=4 y[x]^2 (D[y[x],x])^2 +2(1+3 x)x y[x] D[y[x],x]+3 x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt {-x^3+2 c_1}\\ y(x)&\to \sqrt {-x^3+2 c_1}\\ y(x)&\to -\sqrt {-\frac {x^2}{2}+2 c_1}\\ y(x)&\to \sqrt {-\frac {x^2}{2}+2 c_1} \end{align*}

✓ Sympy. Time used: 0.737 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**3 + x*(6*x + 2)*y(x)*Derivative(y(x), x) + 4*y(x)**2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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