#### 2.382   ODE No. 382

$a x y'(x)-b x^2-c+y'(x)^2=0$ Mathematica : cpu = 0.17664 (sec), leaf count = 201

$\left \{\left \{y(x)\to \frac {1}{2} \left (\frac {1}{2} x \sqrt {a^2 x^2+4 b x^2+4 c}+\frac {2 c \log \left (\sqrt {a^2+4 b} \sqrt {a^2 x^2+4 b x^2+4 c}+a^2 x+4 b x\right )}{\sqrt {a^2+4 b}}-\frac {a x^2}{2}\right )+c_1\right \},\left \{y(x)\to \frac {1}{2} \left (-\frac {1}{2} x \left (\sqrt {x^2 \left (a^2+4 b\right )+4 c}+a x\right )-\frac {2 c \log \left (\sqrt {a^2+4 b} \sqrt {x^2 \left (a^2+4 b\right )+4 c}+a^2 x+4 b x\right )}{\sqrt {a^2+4 b}}\right )+c_1\right \}\right \}$ Maple : cpu = 0.025 (sec), leaf count = 146

$\left \{ y \left ( x \right ) =-{\frac {x}{4}\sqrt { \left ( {a}^{2}+4\,b \right ) {x}^{2}+4\,c}}-{c\ln \left ( \sqrt {{a}^{2}+4\,b}x+\sqrt { \left ( {a}^{2}+4\,b \right ) {x}^{2}+4\,c} \right ) {\frac {1}{\sqrt {{a}^{2}+4\,b}}}}-{\frac {a{x}^{2}}{4}}+{\it \_C1},y \left ( x \right ) ={\frac {x}{4}\sqrt { \left ( {a}^{2}+4\,b \right ) {x}^{2}+4\,c}}+{c\ln \left ( \sqrt {{a}^{2}+4\,b}x+\sqrt { \left ( {a}^{2}+4\,b \right ) {x}^{2}+4\,c} \right ) {\frac {1}{\sqrt {{a}^{2}+4\,b}}}}-{\frac {a{x}^{2}}{4}}+{\it \_C1} \right \}$