\[ -\left (g(x)-f(x)^2\right ) e^{-2 \int _a^x f(\text {xp}) \, d\text {xp}}+2 f(x) y(x) y'(x)+g(x) y(x)^2+y'(x)^2=0 \] ✓ Mathematica : cpu = 0.341279 (sec), leaf count = 89
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]
\[\left \{\left \{y(x)\to e^{-\int _a^x f(K[1]) \, dK[1]} \left (\begin {array}{cc} \{ & \begin {array}{cc} \sin \left (c_1+\int _a^x \sqrt {g(K[1])-f(K[1])^2} \, dK[1]\right ) & g(x)>f(x)^2 \\ \cosh \left (c_1+\int _a^x \sqrt {f(K[1])^2-g(K[1])} \, dK[1]\right ) & g(x)<f(x)^2 \\ c_1 & \text {True} \\\end {array} \\\end {array}\right )\right \}\right \}\] ✓ Maple : cpu = 5.547 (sec), leaf count = 109
dsolve(diff(y(x),x)^2+2*f(x)*y(x)*diff(y(x),x)+g(x)*y(x)^2-(g(x)-f(x)^2)*exp(-2*int(f(xp),xp = a .. x)) = 0,y(x))
\[y \left (x \right ) = \tan \left (-\left (\int {\mathrm e}^{\int _{a}^{x}2 f \left (\mathit {xp} \right )d \mathit {xp}} \sqrt {\left (g \left (x \right )-f \left (x \right )^{2}\right ) {\mathrm e}^{\int _{a}^{x}-4 f \left (\mathit {xp} \right )d \mathit {xp}}}d x \right )+c_{1}\right ) \sqrt {\frac {{\mathrm e}^{-2 \left (\int _{a}^{x}f \left (\mathit {xp} \right )d \mathit {xp} \right )}}{\tan ^{2}\left (-\left (\int {\mathrm e}^{\int _{a}^{x}2 f \left (\mathit {xp} \right )d \mathit {xp}} \sqrt {\left (g \left (x \right )-f \left (x \right )^{2}\right ) {\mathrm e}^{\int _{a}^{x}-4 f \left (\mathit {xp} \right )d \mathit {xp}}}d x \right )+c_{1}\right )+1}}\]