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 = 28.2166 (sec), leaf count = 92

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

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

\[ \left \{ \int _{{\it \_b}}^{y \left ( x \right ) }\!{\frac {{\it \_a}}{f \left ( {{\it \_a}}^{2}+{x}^{2} \right ) }}\,{\rm d}{\it \_a}+\int \!g \left ( x \right ) \,{\rm d}x-{\it \_C1}=0 \right \} \] 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] == 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