\[ -\left (g(x)-f(x)^2\right ) e^{-2 \int _a^x f(\text {xp}) \, d\text {xp}}+2 f(x) y(x) y'(x)+g(x) y(x)^2+y'(x)^2=0 \] ✗ Mathematica : cpu = 56.0079 (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 = 9.915 (sec), leaf count = 164
\[ \left \{ y \left ( x \right ) =-\tan \left ( {\frac {1}{2\,\cos \left ( 2 \right ) +2} \left ( \int \! \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{2}\sqrt {-{\frac { \left ( \left ( f \left ( x \right ) \right ) ^{2}-g \left ( x \right ) \right ) \left ( 4\,\cos \left ( 2 \right ) +\cos \left ( 4 \right ) +3 \right ) }{ \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{4}}}}\,{\rm d}x\sqrt {2}-2\,{\it \_C1}\,\cos \left ( 2 \right ) -2\,{\it \_C1} \right ) } \right ) \sqrt {{{{\rm e}^{-2\,\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \left ( \left ( \tan \left ( {\frac {1}{2\,\cos \left ( 2 \right ) +2} \left ( \int \! \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{2}\sqrt {-{\frac { \left ( \left ( f \left ( x \right ) \right ) ^{2}-g \left ( x \right ) \right ) \left ( 4\,\cos \left ( 2 \right ) +\cos \left ( 4 \right ) +3 \right ) }{ \left ( {{\rm e}^{\int _{a}^{x}\!f \left ( {\it xp} \right ) \,{\rm d}{\it xp}}} \right ) ^{4}}}}\,{\rm d}x\sqrt {2}-2\,{\it \_C1}\,\cos \left ( 2 \right ) -2\,{\it \_C1} \right ) } \right ) \right ) ^{2}+1 \right ) ^{-1}}} \right \} \]