60.7.144 problem 1761 (book 6.170)

Internal problem ID [11694]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1761 (book 6.170)
Date solved : Sunday, March 30, 2025 at 08:42:51 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 112

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

\begin{align*} y &= \frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{-a +1} \int x^{a -1} f \left (x \right )d x +x^{-a +1} c_1 -\int f \left (x \right )d x -c_2 \right )}}{a -1} \\ y &= -\frac {\sqrt {2}\, \sqrt {\left (a -1\right ) \left (x^{-a +1} \int x^{a -1} f \left (x \right )d x +x^{-a +1} c_1 -\int f \left (x \right )d x -c_2 \right )}}{a -1} \\ \end{align*}

✓ Mathematica. Time used: 26.209 (sec). Leaf size: 108

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

\begin{align*} y(x)\to -\sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2} \\ y(x)\to \sqrt {2} \sqrt {\int _1^x-K[2]^{-a} \left (c_1+\int _1^{K[2]}f(K[1]) K[1]^{a-1}dK[1]\right )dK[2]+c_2} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*y(x) + sqrt(a**2*y(x)**2 - 4*x**2*y(x)*Derivative(y(x), (x, 2)) - 4*x*f(x)))/(2*x) cannot be solved by the factorable group method