#### 2.454   ODE No. 454

$a x^2 y'(x)^2-(a-1) a x^2-2 a x y(x) y'(x)+y(x)^2=0$ Mathematica : cpu = 0.37691 (sec), leaf count = 241

$\left \{\left \{y(x)\to -\frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right )\right \},\left \{y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-x^{2 \sqrt {\frac {a-1}{a}}}+e^{2 c_1}\right )\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right )\right \},\left \{y(x)\to \frac {1}{2} \sqrt {a} e^{-c_1} x^{1-\sqrt {\frac {a-1}{a}}} \left (-1+e^{2 c_1} x^{2 \sqrt {\frac {a-1}{a}}}\right )\right \}\right \}$ Maple : cpu = 0.219 (sec), leaf count = 106

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