2.582   ODE No. 582

\[ y'(x)=\frac {a x^2 F\left (\frac {a x y(x)+1}{a x}\right )+1}{a x^2} \]

✓ Mathematica : cpu = 0.285467 (sec), leaf count = 142

DSolve[Derivative[1][y][x] == (1 + a*x^2*F[(1 + a*x*y[x])/(a*x)])/(a*x^2),y[x],x]
 

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

✓ Maple : cpu = 0.178 (sec), leaf count = 30

dsolve(diff(y(x),x) = (1+F((a*x*y(x)+1)/a/x)*a*x^2)/a/x^2,y(x))
 

\[y \left (x \right ) = \frac {\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\frac {1}{F \left (\textit {\_a} \right )}d \textit {\_a} +c_{1} \right ) x a -1}{x a}\]