60.1.249 problem 254

Internal problem ID [10263]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 254
Date solved : Wednesday, March 05, 2025 at 08:50:28 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class C`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 59

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

\begin{align*} y &= \frac {-1+\sqrt {1-4 \ln \left (x \right )+4 c_{1}}}{2 x \left (-\ln \left (x \right )+c_{1} \right )} \\ y &= \frac {1+\sqrt {1-4 \ln \left (x \right )+4 c_{1}}}{2 \left (\ln \left (x \right )-c_{1} \right ) x} \\ \end{align*}

✓ Mathematica. Time used: 1.188 (sec). Leaf size: 86

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

\begin{align*} y(x)\to \frac {2}{x+\sqrt {-\frac {1}{x^3}} x^2 \sqrt {-x (-4 \log (x)+1+4 c_1)}} \\ y(x)\to \frac {2}{x+\left (-\frac {1}{x^3}\right )^{3/2} x^5 \sqrt {-x (-4 \log (x)+1+4 c_1)}} \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.011 (sec). Leaf size: 51

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

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