2.5.20 problem 20

Internal problem ID [748]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 04:08:26 AM
CAS classification : [_separable]

\begin{align*} 2 x y^{3}+y^{2} y^{\prime }&=6 x \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 64

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

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

✓ Mathematica. Time used: 1.794 (sec). Leaf size: 115

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

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

✓ Sympy. Time used: 1.506 (sec). Leaf size: 68

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

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