69.7.18 problem 193

Internal problem ID [18098]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 193
Date solved : Thursday, October 02, 2025 at 02:42:05 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 82

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

\begin{align*} y &= \left (-x^{2} \left (x \ln \left (x \right )-c_1 -x \right )\right )^{{1}/{3}} \\ y &= -\frac {\left (-x^{2} \left (x \ln \left (x \right )-c_1 -x \right )\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (-x^{2} \left (x \ln \left (x \right )-c_1 -x \right )\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.543 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to \sqrt [3]{x^2 (x+x (-\log (x))+c_1)}\\ y(x)&\to -\sqrt [3]{-1} \sqrt [3]{x^2 (x+x (-\log (x))+c_1)}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{x^2 (x+x (-\log (x))+c_1)} \end{align*}

✓ Sympy. Time used: 1.107 (sec). Leaf size: 73

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

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