\[ -\left (g(x)-f(x)^2\right ) e^{-2 \int _a^x f(\text {xp}) \, d\text {xp}}+2 f(x) y(x) y'(x)+g(x) y(x)^2+y'(x)^2=0 \] ✗ Mathematica : cpu = 88.2118 (sec), leaf count = 0 , could not solve
DSolve[-((-f[x]^2 + g[x])/E^(2*Integrate[f[xp], {xp, a, x}])) + g[x]*y[x]^2 + 2*f[x]*y[x]*Derivative[1][y][x] + Derivative[1][y][x]^2 == 0, y[x], x]
✓ Maple : cpu = 10.447 (sec), leaf count = 109
\[ \left \{ y \left ( x \right ) =\tan \left ( -\int \! \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{2}\sqrt {{\frac {g \left ( x \right ) - \left ( f \left ( x \right ) \right ) ^{2}}{ \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{4}}}}\,{\rm d}x+{\it \_C1} \right ) \sqrt {{{{\rm e}^{-2\,\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \left ( \left ( \tan \left ( -\int \! \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{2}\sqrt {{\frac {g \left ( x \right ) - \left ( f \left ( x \right ) \right ) ^{2}}{ \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{4}}}}\,{\rm d}x+{\it \_C1} \right ) \right ) ^{2}+1 \right ) ^{-1}}} \right \} \]