78.8.20 problem 20

Internal problem ID [18139]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 20
Date solved : Thursday, March 13, 2025 at 11:40:59 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 59

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

\begin{align*} y &= \frac {c_{1} x^{2}-\sqrt {2 c_{1}^{2} x^{4}+1}}{c_{1} x} \\ y &= \frac {c_{1} x^{2}+\sqrt {2 c_{1}^{2} x^{4}+1}}{c_{1} x} \\ \end{align*}

✓ Mathematica. Time used: 0.697 (sec). Leaf size: 99

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

\begin{align*} y(x)\to x-\frac {\sqrt {2 x^4+e^{2 c_1}}}{x} \\ y(x)\to x+\frac {\sqrt {2 x^4+e^{2 c_1}}}{x} \\ y(x)\to x-\frac {\sqrt {2} \sqrt {x^4}}{x} \\ y(x)\to \frac {\sqrt {2} \sqrt {x^4}}{x}+x \\ \end{align*}

✓ Sympy. Time used: 1.366 (sec). Leaf size: 31

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

\[ \left [ y{\left (x \right )} = x - \frac {\sqrt {C_{1} + 2 x^{4}}}{x}, \ y{\left (x \right )} = x + \frac {\sqrt {C_{1} + 2 x^{4}}}{x}\right ] \]