##### 4.9.34 $$g(x) f\left (x^2+y(x)^2\right )+y(x) y'(x)+x=0$$

ODE
$g(x) f\left (x^2+y(x)^2\right )+y(x) y'(x)+x=0$ ODE Classiﬁcation

[NONE]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 0.564775 (sec), leaf count = 95

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

Maple
cpu = 0.16 (sec), leaf count = 30

$\left [\int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}}{f \left (\textit {\_a}^{2}+x^{2}\right )}d \textit {\_a} +\int g \left (x \right )d x -\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[x + f[x^2 + y[x]^2]*g[x] + y[x]*y'[x] == 0,y[x],x]

Mathematica raw output

Solve[C[1] == Inactive[Integrate][g[K[1]] + K[1]/f[K[1]^2 + y[x]^2], {K[1], 1, x
}] + Inactive[Integrate][K[2]/f[x^2 + K[2]^2] - Inactive[Integrate][(-2*K[1]*K[2
]*Derivative[1][f][K[1]^2 + K[2]^2])/f[K[1]^2 + K[2]^2]^2, {K[1], 1, x}], {K[2],
 1, y[x]}], y[x]]

Maple raw input

dsolve(y(x)*diff(y(x),x)+x+f(x^2+y(x)^2)*g(x) = 0, y(x))

Maple raw output

[Int(1/f(_a^2+x^2)*_a,_a = _b .. y(x))+Int(g(x),x)-_C1 = 0]