\[ \boxed { \left ( \left ( y \left ( x \right ) \right ) ^{4}-{a}^{2}{x}^{2} \right ) \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+2\,{a}^{2}xy \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( y \left ( x \right ) \right ) ^{2} \left ( \left ( y \left ( x \right ) \right ) ^{2}-{a}^{2} \right ) =0} \]
Mathematica: cpu = 29.710773 (sec), leaf count = 443 \[ \left \{\text {Solve}\left [\left \{x=\frac {a^2 \text {K$\$$1432838} y(\text {K$\$$1432838})-\sqrt {a^2 \text {K$\$$1432838}^2 \left (\text {K$\$$1432838}^2+1\right ) y(\text {K$\$$1432838})^4}}{a^2 \text {K$\$$1432838}^2},y(x)=\frac {1}{4} \left (-\frac {a c_1 \sqrt {\text {K$\$$1432838}}}{\sqrt [4]{\text {K$\$$1432838}^2+1}}-\frac {2 a \sqrt [4]{\frac {\text {K$\$$1432838}^2+1}{\text {K$\$$1432838}^2}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {1}{\text {K$\$$1432838}^2}\right )}{\sqrt {\text {K$\$$1432838}^2+1}}\right )\right \},\{y(x),\text {K$\$$1432838}\}\right ],\text {Solve}\left [\left \{x=\frac {a^2 \text {K$\$$1432838} y(\text {K$\$$1432838})-\sqrt {a^2 \text {K$\$$1432838}^2 \left (\text {K$\$$1432838}^2+1\right ) y(\text {K$\$$1432838})^4}}{a^2 \text {K$\$$1432838}^2},y(x)=\frac {1}{4} \left (\frac {2 a \sqrt [4]{\frac {\text {K$\$$1432838}^2+1}{\text {K$\$$1432838}^2}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {1}{\text {K$\$$1432838}^2}\right )}{\sqrt {\text {K$\$$1432838}^2+1}}-\frac {a c_1 \sqrt {\text {K$\$$1432838}}}{\sqrt [4]{\text {K$\$$1432838}^2+1}}\right )\right \},\{y(x),\text {K$\$$1432838}\}\right ],\text {Solve}\left [\left \{x=\frac {\sqrt {a^2 \text {K$\$$1435339}^2 \left (\text {K$\$$1435339}^2+1\right ) y(\text {K$\$$1435339})^4}+a^2 \text {K$\$$1435339} y(\text {K$\$$1435339})}{a^2 \text {K$\$$1435339}^2},y(x)=\frac {1}{4} \left (\frac {a c_1 \sqrt {\text {K$\$$1435339}}}{\sqrt [4]{\text {K$\$$1435339}^2+1}}-\frac {2 a \sqrt [4]{\frac {\text {K$\$$1435339}^2+1}{\text {K$\$$1435339}^2}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {1}{\text {K$\$$1435339}^2}\right )}{\sqrt {\text {K$\$$1435339}^2+1}}\right )\right \},\{y(x),\text {K$\$$1435339}\}\right ],\text {Solve}\left [\left \{x=\frac {\sqrt {a^2 \text {K$\$$1435339}^2 \left (\text {K$\$$1435339}^2+1\right ) y(\text {K$\$$1435339})^4}+a^2 \text {K$\$$1435339} y(\text {K$\$$1435339})}{a^2 \text {K$\$$1435339}^2},y(x)=\frac {1}{4} \left (\frac {a c_1 \sqrt {\text {K$\$$1435339}}}{\sqrt [4]{\text {K$\$$1435339}^2+1}}+\frac {2 a \sqrt [4]{\frac {\text {K$\$$1435339}^2+1}{\text {K$\$$1435339}^2}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {1}{\text {K$\$$1435339}^2}\right )}{\sqrt {\text {K$\$$1435339}^2+1}}\right )\right \},\{y(x),\text {K$\$$1435339}\}\right ]\right \} \]
Maple: cpu = 0 (sec), leaf count = 0 \[ \text {hanged} \]