83.10.12 problem 12

Internal problem ID [21981]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IV. First order differential equations of higher degree. Ex. XI at page 69
Problem number : 12
Date solved : Thursday, October 02, 2025 at 08:21:23 PM
CAS classification : [_quadrature]

\begin{align*} \left (x^{2}-2 y x \right ) {y^{\prime }}^{2}-\left (3 x^{2}+2 y\right ) \left (x -2 y\right ) y^{\prime }+6 x y \left (x -2 y\right )&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

ode:=(x^2-2*x*y(x))*diff(y(x),x)^2-(3*x^2+2*y(x))*(x-2*y(x))*diff(y(x),x)+6*x*y(x)*(x-2*y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x}{2} \\ y &= c_1 \,x^{2} \\ y &= \frac {3 x^{2}}{2}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 54

ode=(x^2-2*x*y[x])* D[y[x],x]^2 - (3*x^2+2*y[x])*(x-2*y[x])*D[y[x],x] + 6*x*y[x]*(x-2*y[x])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x}{2}\\ y(x)&\to c_1 x^2\\ y(x)&\to \frac {3 x^2}{2}+c_1\\ y(x)&\to \frac {x}{2}\\ y(x)&\to \frac {3 x^2}{2} \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 22

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

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