83.9.8 problem 8

Internal problem ID [21966]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. X at page 57
Problem number : 8
Date solved : Thursday, October 02, 2025 at 08:20:16 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.117 (sec). Leaf size: 63

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

\begin{align*} y &= \frac {-c_1 x -\sqrt {-3 c_1^{2} x^{2}+10 c_1 x +1}+1}{2 c_1} \\ y &= \frac {-c_1 x +1+\sqrt {-3 c_1^{2} x^{2}+10 c_1 x +1}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 1.49 (sec). Leaf size: 81

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

\begin{align*} y(x)&\to \frac {1}{2} \left (-\sqrt {-3 x^2+10 e^{c_1} x+e^{2 c_1}}-x+e^{c_1}\right )\\ y(x)&\to \frac {1}{2} \left (\sqrt {-3 x^2+10 e^{c_1} x+e^{2 c_1}}-x+e^{c_1}\right ) \end{align*}

✓ Sympy. Time used: 1.561 (sec). Leaf size: 65

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

\[ \left [ y{\left (x \right )} = - \frac {x}{2} - \frac {\sqrt {- 3 x^{2} + 10 x e^{C_{1}} + e^{2 C_{1}}}}{2} + \frac {e^{C_{1}}}{2}, \ y{\left (x \right )} = - \frac {x}{2} + \frac {\sqrt {- 3 x^{2} + 10 x e^{C_{1}} + e^{2 C_{1}}}}{2} + \frac {e^{C_{1}}}{2}\right ] \]