#### 2.477   ODE No. 477

$a y(x) y'(x)^2+(2 x-b) y'(x)-y(x)=0$ Mathematica : cpu = 0.214434 (sec), leaf count = 146

$\left \{\left \{y(x)\to -\frac {e^{\frac {c_1}{2}} \sqrt {2 b-4 x+e^{c_1}}}{2 \sqrt {a}}\right \},\left \{y(x)\to \frac {e^{\frac {c_1}{2}} \sqrt {2 b-4 x+e^{c_1}}}{2 \sqrt {a}}\right \},\left \{y(x)\to -\sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}-b+2 x}\right \},\left \{y(x)\to \sqrt {2} e^{\frac {c_1}{2}} \sqrt {2 a e^{c_1}-b+2 x}\right \}\right \}$ Maple : cpu = 0.643 (sec), leaf count = 622

$\left \{ \int _{{\it \_b}}^{x}\!{ \left ( -4\,{\it \_a}+2\,b-2\,\sqrt {4\,a \left ( y \left ( x \right ) \right ) ^{2}+ \left ( b-2\,{\it \_a} \right ) ^{2}} \right ) \left ( \left ( -b+2\,{\it \_a} \right ) \sqrt {4\,a \left ( y \left ( x \right ) \right ) ^{2}+ \left ( b-2\,{\it \_a} \right ) ^{2}}+4\,a \left ( y \left ( x \right ) \right ) ^{2}+ \left ( b-2\,{\it \_a} \right ) ^{2} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!4\,{\frac {a{\it \_f}}{-4\,a{{\it \_f}}^{2}+\sqrt {4\,a{{\it \_f}}^{2}+{b}^{2}-4\,bx+4\,{x}^{2}}b-2\,\sqrt {4\,a{{\it \_f}}^{2}+{b}^{2}-4\,bx+4\,{x}^{2}}x-{b}^{2}+4\,bx-4\,{x}^{2}}}-\int _{{\it \_b}}^{x}\!{ \left ( -32\,{\frac {{a}^{2}{{\it \_f}}^{3}}{\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}}}}-16\,a{\it \_f}\, \left ( -2\,{\it \_a}+b-\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}} \right ) \right ) \left ( 4\,a{{\it \_f}}^{2}+2\,\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}}{\it \_a}-\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}}b+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,\int _{{\it \_b}}^{x}\!{ \left ( -4\,{\it \_a}+2\,b+2\,\sqrt {4\,a \left ( y \left ( x \right ) \right ) ^{2}+ \left ( b-2\,{\it \_a} \right ) ^{2}} \right ) \left ( \left ( b-2\,{\it \_a} \right ) \sqrt {4\,a \left ( y \left ( x \right ) \right ) ^{2}+ \left ( b-2\,{\it \_a} \right ) ^{2}}+4\,a \left ( y \left ( x \right ) \right ) ^{2}+ \left ( b-2\,{\it \_a} \right ) ^{2} \right ) ^{-1}}\,{\rm d}{\it \_a}+\int ^{y \left ( x \right ) }\!-4\,{\frac {a{\it \_f}}{4\,a{{\it \_f}}^{2}+\sqrt {4\,a{{\it \_f}}^{2}+{b}^{2}-4\,bx+4\,{x}^{2}}b-2\,\sqrt {4\,a{{\it \_f}}^{2}+{b}^{2}-4\,bx+4\,{x}^{2}}x+{b}^{2}-4\,bx+4\,{x}^{2}}}-\int _{{\it \_b}}^{x}\!{ \left ( 32\,{\frac {{a}^{2}{{\it \_f}}^{3}}{\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}}}}-16\,a{\it \_f}\, \left ( -2\,{\it \_a}+b+\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}} \right ) \right ) \left ( 4\,a{{\it \_f}}^{2}+\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}}b-2\,\sqrt {4\,a{{\it \_f}}^{2}+4\,{{\it \_a}}^{2}-4\,{\it \_a}\,b+{b}^{2}}{\it \_a}+{b}^{2}-4\,{\it \_a}\,b+4\,{{\it \_a}}^{2} \right ) ^{-2}}\,{\rm d}{\it \_a}{d{\it \_f}}+{\it \_C1}=0,y \left ( x \right ) =-{\frac {-2\,x+b}{2}{\frac {1}{\sqrt {-a}}}},y \left ( x \right ) ={\frac {-2\,x+b}{2}{\frac {1}{\sqrt {-a}}}} \right \}$