#### 2.549   ODE No. 549

$x^2 \left (y'(x)^2+1\right )^3-a^2=0$ Mathematica : cpu = 0.256686 (sec), leaf count = 216

$\left \{\left \{y(x)\to c_1-x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}\right \},\left \{y(x)\to x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1\right \},\left \{y(x)\to c_1-x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}\right \},\left \{y(x)\to c_1+x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}\right \},\left \{y(x)\to c_1-x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}\right \},\left \{y(x)\to c_1+x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}\right \}\right \}$ Maple : cpu = 0.348 (sec), leaf count = 553

$\left \{ y \left ( x \right ) ={1\sqrt {-{\frac {1}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-{a}^{2} \right ) }} \left ( {a}^{2}- \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}} \right ) \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={1\sqrt {-{\frac {1}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-{a}^{2} \right ) }} \left ( \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}-{a}^{2} \right ) \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={-{\frac {i}{4}}\sqrt {2} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+ \left ( \sqrt {3}+i \right ) {a}^{2} \right ) \sqrt {{\frac {i}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2}+\sqrt {3}{a}^{2} \right ) }} \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={{\frac {i}{4}}\sqrt {2} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+ \left ( \sqrt {3}+i \right ) {a}^{2} \right ) \sqrt {{\frac {i}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2}+\sqrt {3}{a}^{2} \right ) }} \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={-{\frac {i}{4}}\sqrt {2}\sqrt {i \left ( i\sqrt [3]{{a}^{2}x}-\sqrt {3}\sqrt [3]{{a}^{2}x}+2\,ix \right ) x}\sqrt {{\frac {1}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2}-\sqrt {3}{a}^{2} \right ) }} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2}-\sqrt {3}{a}^{2} \right ) {\frac {1}{\sqrt { \left ( i\sqrt [3]{{a}^{2}x}-\sqrt {3}\sqrt [3]{{a}^{2}x}+2\,ix \right ) x}}} \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1},y \left ( x \right ) ={{\frac {i}{4}}\sqrt {2}\sqrt {i \left ( i\sqrt [3]{{a}^{2}x}-\sqrt {3}\sqrt [3]{{a}^{2}x}+2\,ix \right ) x}\sqrt {{\frac {1}{{a}^{4}} \left ( {a}^{2}x \right ) ^{{\frac {4}{3}}} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2}-\sqrt {3}{a}^{2} \right ) }} \left ( 2\,i \left ( {a}^{2}x \right ) ^{{\frac {2}{3}}}+i{a}^{2}-\sqrt {3}{a}^{2} \right ) {\frac {1}{\sqrt { \left ( i\sqrt [3]{{a}^{2}x}-\sqrt {3}\sqrt [3]{{a}^{2}x}+2\,ix \right ) x}}} \left ( {a}^{2}x \right ) ^{-{\frac {2}{3}}}}+{\it \_C1} \right \}$