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