#### 2.478   ODE No. 478

$\left (y'(x)^2+1\right ) (a y(x)+b)-c=0$ Mathematica : cpu = 0.31172 (sec), leaf count = 223

$\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\frac {c \sqrt {-a c} \sqrt {\frac {\text {\#1} a+b}{c}} \sin ^{-1}\left (\frac {a \sqrt {-\text {\#1} a-b+c}}{\sqrt {-a} \sqrt {-a c}}\right )}{\sqrt {-a}}-(\text {\#1} a+b) \sqrt {-\text {\#1} a-b+c}}{a \sqrt {\text {\#1} a+b}}\& \right ][-x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\frac {c \sqrt {-a c} \sqrt {\frac {\text {\#1} a+b}{c}} \sin ^{-1}\left (\frac {a \sqrt {-\text {\#1} a-b+c}}{\sqrt {-a} \sqrt {-a c}}\right )}{\sqrt {-a}}-(\text {\#1} a+b) \sqrt {-\text {\#1} a-b+c}}{a \sqrt {\text {\#1} a+b}}\& \right ][x+c_1]\right \}\right \}$ Maple : cpu = 0.318 (sec), leaf count = 88

$\left \{ x-\int ^{y \left ( x \right ) }\!{({\it \_a}\,a+b){\frac {1}{\sqrt {- \left ( {\it \_a}\,a+b \right ) \left ( {\it \_a}\,a+b-c \right ) }}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!-{({\it \_a}\,a+b){\frac {1}{\sqrt {- \left ( {\it \_a}\,a+b \right ) \left ( {\it \_a}\,a+b-c \right ) }}}}{d{\it \_a}}-{\it \_C1}=0,y \left ( x \right ) ={\frac {-b+c}{a}} \right \}$