60.7.62 problem 1670 (book 6.79)

Internal problem ID [11612]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1670 (book 6.79)
Date solved : Sunday, March 30, 2025 at 08:31:33 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 32

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

\[ y = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_2 -\int _{}^{\textit {\_Z}}\frac {1}{-2 \textit {\_f} -1+{\mathrm e}^{\textit {\_f}} c_1}d \textit {\_f} \right )}{x} \]

✓ Mathematica. Time used: 0.343 (sec). Leaf size: 326

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

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {x}{e^{x K[2]} c_1+x K[2]+e^{x K[2]} \int _1^{x K[2]}-e^{-K[1]} K[1]dK[1]}dK[2]-\int _1^x\left (\int _1^{y(x)}\left (\frac {K[3] \left (-K[3] K[2]^2+e^{K[2] K[3]} c_1 K[2]+e^{K[2] K[3]} \int _1^{K[2] K[3]}-e^{-K[1]} K[1]dK[1] K[2]+K[2]\right )}{\left (e^{K[2] K[3]} c_1+K[2] K[3]+e^{K[2] K[3]} \int _1^{K[2] K[3]}-e^{-K[1]} K[1]dK[1]\right ){}^2}-\frac {1}{e^{K[2] K[3]} c_1+K[2] K[3]+e^{K[2] K[3]} \int _1^{K[2] K[3]}-e^{-K[1]} K[1]dK[1]}\right )dK[2]-\frac {e^{K[3] y(x)} c_1+e^{K[3] y(x)} \int _1^{K[3] y(x)}-e^{-K[1]} K[1]dK[1]}{K[3] \left (e^{K[3] y(x)} c_1+K[3] y(x)+e^{K[3] y(x)} \int _1^{K[3] y(x)}-e^{-K[1]} K[1]dK[1]\right )}\right )dK[3]=c_2,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(x**3*Derivative(y(x), (x, 2)) + x**2*y(x)**2 + 1) + 1)/x**2 cannot be solved by the factorable group method