15.2.20 problem 20

Internal problem ID [2890]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 6, page 25
Problem number : 20
Date solved : Tuesday, March 04, 2025 at 03:00:58 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

With initial conditions

\begin{align*} y \left (1\right )&=-1 \end{align*}

✓ Maple. Time used: 0.336 (sec). Leaf size: 111

ode:=(3*x*y(x)-2*x^2)*diff(y(x),x) = 2*y(x)^2-x*y(x); 
ic:=y(1) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (-27 x^{2}+x^{3}+3 \sqrt {3}\, \sqrt {-x^{4} \left (2 x -27\right )}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{6}-\frac {x \left (i \sqrt {3}\, x +x -2 \left (-27 x^{2}+x^{3}+3 \sqrt {3}\, \sqrt {-x^{4} \left (2 x -27\right )}\right )^{{1}/{3}}\right )}{6 \left (-27 x^{2}+x^{3}+3 \sqrt {3}\, \sqrt {-x^{4} \left (2 x -27\right )}\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 60.341 (sec). Leaf size: 140

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

\[ y(x)\to \frac {\left (\sqrt [3]{3 \sqrt {3} \sqrt {-x^4 (2 x-27)}+x^3-27 x^2}-x\right ) \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{3 \sqrt {3} \sqrt {-x^4 (2 x-27)}+x^3-27 x^2}+i \sqrt {3} x+x\right )}{6 \sqrt [3]{3 \sqrt {3} \sqrt {-x^4 (2 x-27)}+x^3-27 x^2}} \]

✓ Sympy. Time used: 41.778 (sec). Leaf size: 87

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

\[ y{\left (x \right )} = - \frac {\sqrt [3]{2} x^{2}}{3 \sqrt [3]{- 2 x^{3} + 54 x^{2} + 3 \sqrt {6} \sqrt {x^{4} \left (54 - 4 x\right )}}} + \frac {x}{3} - \frac {2^{\frac {2}{3}} \sqrt [3]{- 2 x^{3} + 54 x^{2} + 3 \sqrt {6} \sqrt {x^{4} \left (54 - 4 x\right )}}}{6} \]