60.1.424 problem 435

Internal problem ID [10438]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 435
Date solved : Sunday, March 30, 2025 at 04:43:18 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} x^{2} {y^{\prime }}^{2}-2 x y y^{\prime }+y \left (y+1\right )-x&=0 \end{align*}

✓ Maple. Time used: 0.062 (sec). Leaf size: 22

ode:=x^2*diff(y(x),x)^2-2*x*y(x)*diff(y(x),x)+y(x)*(1+y(x))-x = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.118 (sec). Leaf size: 55

ode=-x + y[x]*(1 + 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 x+\frac {c_1{}^2 x}{4}-i c_1 \sqrt {x}-1 \\ y(x)\to x+\frac {c_1{}^2 x}{4}+i c_1 \sqrt {x}-1 \\ \end{align*}

✓ Sympy. Time used: 21.505 (sec). Leaf size: 95

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x)**2 - 2*x*y(x)*Derivative(y(x), x) - x + (y(x) + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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