60.1.432 problem 444

Internal problem ID [10446]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 444
Date solved : Wednesday, March 05, 2025 at 10:49:35 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 1.269 (sec). Leaf size: 124

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

\begin{align*} y &= 4 x \\ y &= 0 \\ y &= -\frac {2 c_{1}^{2} \left (-c_{1} \sqrt {2}+x \right )}{-2 c_{1}^{2}+x^{2}} \\ y &= -\frac {2 c_{1}^{2} \left (c_{1} \sqrt {2}+x \right )}{-2 c_{1}^{2}+x^{2}} \\ y &= \frac {c_{1}^{3} \sqrt {2}-2 c_{1}^{2} x}{-2 c_{1}^{2}+4 x^{2}} \\ y &= \frac {c_{1}^{2} \left (c_{1} \sqrt {2}+2 x \right )}{2 c_{1}^{2}-4 x^{2}} \\ \end{align*}

✓ Mathematica. Time used: 1.275 (sec). Leaf size: 61

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

\begin{align*} y(x)\to -\frac {4 e^{-2 c_1}}{-2+e^{2 c_1} x} \\ y(x)\to -\frac {4 e^{-2 c_1}}{2+e^{2 c_1} x} \\ y(x)\to 0 \\ y(x)\to 4 x \\ \end{align*}

✓ Sympy. Time used: 11.242 (sec). Leaf size: 17

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

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