76.13.5 problem Ex. 5

Internal problem ID [20090]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter III. Equations of the first order but not of the first degree. Problems at page 32
Problem number : Ex. 5
Date solved : Thursday, October 02, 2025 at 05:22:37 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: 0.006 (sec). Leaf size: 59

ode:=4*y(x)^2*diff(y(x),x)^2+2*diff(y(x),x)*x*y(x)*(3*x+1)+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: 0.111 (sec). Leaf size: 81

ode=4*y[x]^2*D[y[x],x]^2+2*D[y[x],x]*x*y[x]*(3*x+1)+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: 0.730 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**3 + 2*x*(3*x + 1)*y(x)*Derivative(y(x), x) + 4*y(x)**2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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