4.9.34 $g(x)f(x^2+y(x{)}^2)+y(x)y^{\prime }(x)+x=0$

ODE
$$g(x)f(x^2+y(x{)}^2)+y(x)y^{\prime }(x)+x=0$$ ODE Classification

[NONE]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 28.2166 (sec), leaf count = 92

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

Maple ✓
cpu = 0.125 (sec), leaf count = 30

$$\{{\int }_{\mathit{\_}\mathit{b}}^{y\left(x\right)}\frac{\mathit{\_}\mathit{a}}{f({\mathit{\_}\mathit{a}}^2+x^2)}\mathrm{d}\mathit{\_}\mathit{a}+\int g\left(x\right)\mathrm{d}x-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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] == Integrate[g[K[1]] + K[1]/f[K[1]^2 + y[x]^2], {K[1], 1, x}] + Integ
rate[-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]/f[x^2 + K[2]^2], {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),'implicit')

Maple raw output

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