32.6.29 problem Exercise 12.29, page 103

Internal problem ID [5894]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.29, page 103
Date solved : Sunday, March 30, 2025 at 10:24:14 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 59

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

\begin{align*} y &= \frac {c_1 \,x^{2}-\sqrt {2 x^{4} c_1^{2}+1}}{c_1 x} \\ y &= \frac {c_1 \,x^{2}+\sqrt {2 x^{4} c_1^{2}+1}}{c_1 x} \\ \end{align*}

✓ Mathematica. Time used: 0.749 (sec). Leaf size: 99

ode=(x*y[x]-x^2)*D[y[x],x]+y[x]^2-3*x*y[x]-2*x^2==0; 
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: 1.271 (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)
 

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