84.14.15 problem 7.23

Internal problem ID [22172]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 7. Integrating factors. Supplementary problems
Problem number : 7.23
Date solved : Thursday, October 02, 2025 at 08:33:22 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 41

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

\begin{align*} y &= 0 \\ y &= \frac {{\mathrm e}^{-\frac {3 c_1}{2}} 2^{{1}/{3}}}{x^{{3}/{2}} \left (\frac {{\mathrm e}^{-\frac {9 c_1}{2}}}{x^{{9}/{2}} \operatorname {LambertW}\left (\frac {{\mathrm e}^{-\frac {9 c_1}{2}}}{2 x^{{9}/{2}}}\right )}\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 11.461 (sec). Leaf size: 221

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

\begin{align*} y(x)&\to 0\\ y(x)&\to -\sqrt [3]{-2} \sqrt [3]{W\left (-\frac {1}{2} \sqrt {\frac {e^{3 c_1}}{x^9}}\right )}\\ y(x)&\to \sqrt [3]{2} \sqrt [3]{W\left (-\frac {1}{2} \sqrt {\frac {e^{3 c_1}}{x^9}}\right )}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{2} \sqrt [3]{W\left (-\frac {1}{2} \sqrt {\frac {e^{3 c_1}}{x^9}}\right )}\\ y(x)&\to -\sqrt [3]{-2} \sqrt [3]{W\left (\frac {1}{2} \sqrt {\frac {e^{3 c_1}}{x^9}}\right )}\\ y(x)&\to \sqrt [3]{2} \sqrt [3]{W\left (\frac {1}{2} \sqrt {\frac {e^{3 c_1}}{x^9}}\right )}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{2} \sqrt [3]{W\left (\frac {1}{2} \sqrt {\frac {e^{3 c_1}}{x^9}}\right )}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.319 (sec). Leaf size: 22

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

\[ \left [ \frac {y^{3}{\left (x \right )}}{3} + 3 \log {\left (x \right )} + 2 \log {\left (y{\left (x \right )} \right )} = C_{1}, \ y{\left (x \right )} = 0\right ] \]