#### 2.502   ODE No. 502

$(a y(x)-b x)^2 \left (a^2 y'(x)^2+b^2\right )-c^2 \left (a y'(x)+b\right )^2=0$ Mathematica : cpu = 1.19749 (sec), leaf count = 100

$\left \{\left \{y(x)\to \frac {b c_1}{a}-\frac {\sqrt {2 b^2 c_1 x-b^2 c_1{}^2+b^2 \left (-x^2\right )+c^2}}{a}\right \},\left \{y(x)\to \frac {\sqrt {2 b^2 c_1 x-b^2 c_1{}^2+b^2 \left (-x^2\right )+c^2}}{a}+\frac {b c_1}{a}\right \}\right \}$ Maple : cpu = 0.69 (sec), leaf count = 195

$\left \{ y \left ( x \right ) ={\frac {bx-\sqrt {2}c}{a}},y \left ( x \right ) ={\frac {bx+\sqrt {2}c}{a}},y \left ( x \right ) ={\frac {1}{a} \left ( {\it RootOf} \left ( -x+\int ^{{\it \_Z}}\!{\frac {a}{ \left ( 2\,{{\it \_a}}^{2}{a}^{2}-4\,{c}^{2} \right ) b} \left ( -{{\it \_a}}^{2}{a}^{2}+2\,{c}^{2}+\sqrt {-{a}^{2}{{\it \_a}}^{2} \left ( {{\it \_a}}^{2}{a}^{2}-2\,{c}^{2} \right ) } \right ) }{d{\it \_a}}+{\it \_C1} \right ) a+bx \right ) },y \left ( x \right ) ={\frac {1}{a} \left ( {\it RootOf} \left ( -x+\int ^{{\it \_Z}}\!-{\frac {a}{ \left ( 2\,{{\it \_a}}^{2}{a}^{2}-4\,{c}^{2} \right ) b} \left ( {{\it \_a}}^{2}{a}^{2}-2\,{c}^{2}+\sqrt {-{a}^{2}{{\it \_a}}^{2} \left ( {{\it \_a}}^{2}{a}^{2}-2\,{c}^{2} \right ) } \right ) }{d{\it \_a}}+{\it \_C1} \right ) a+bx \right ) } \right \}$