4.21.19 \(y'(x)^2 \left (a^2 r(x,y(x))-x^2\right )+a^2 r(x,y(x))+2 x y(x) y'(x)-y(x)^2=0\)

ODE
\[ y'(x)^2 \left (a^2 r(x,y(x))-x^2\right )+a^2 r(x,y(x))+2 x y(x) y'(x)-y(x)^2=0 \] ODE Classification

[NONE]

Book solution method
Change of variable, polar coordinates

Mathematica
cpu = 51.7921 (sec), leaf count = 0 , could not solve

DSolve[a^2*r[x, y[x]] - y[x]^2 + 2*x*y[x]*Derivative[1][y][x] + (-x^2 + a^2*r[x, y[x]])*Derivative[1][y][x]^2 == 0, y[x], x]

Maple
cpu = 4.813 (sec), leaf count = 0 , could not solve

dsolve((a^2*r(x,y(x))-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a^2*r(x,y(x))-y(x)^2 = 0, y(x),'implicit')

Mathematica raw input

DSolve[a^2*r[x, y[x]] - y[x]^2 + 2*x*y[x]*y'[x] + (-x^2 + a^2*r[x, y[x]])*y'[x]^2 == 0,y[x],x]

Mathematica raw output

DSolve[a^2*r[x, y[x]] - y[x]^2 + 2*x*y[x]*Derivative[1][y][x] + (-x^2 + a^2*r[x,
 y[x]])*Derivative[1][y][x]^2 == 0, y[x], x]

Maple raw input

dsolve((a^2*r(x,y(x))-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a^2*r(x,y(x))-y(x)^2 = 0, y(x),'implicit')

Maple raw output

dsolve((a^2*r(x,y(x))-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+a^2*r(x,y(x))-y(
x)^2 = 0, y(x),'implicit')