2.478   ODE No. 478

\[ \left (y'(x)^2+1\right ) (a y(x)+b)-c=0 \]

✓ Mathematica : cpu = 0.133722 (sec), leaf count = 141

DSolve[-c + (b + a*y[x])*(1 + Derivative[1][y][x]^2) == 0,y[x],x]
 

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {c \tan ^{-1}\left (\frac {\sqrt {\text {$\#$1} a+b}}{\sqrt {-\text {$\#$1} a-b+c}}\right )-\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\& \right ][-x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {c \tan ^{-1}\left (\frac {\sqrt {\text {$\#$1} a+b}}{\sqrt {-\text {$\#$1} a-b+c}}\right )-\sqrt {\text {$\#$1} a+b} \sqrt {-\text {$\#$1} a-b+c}}{a}\& \right ][x+c_1]\right \}\right \}\]

✓ Maple : cpu = 0.5 (sec), leaf count = 211

dsolve((a*y(x)+b)*(diff(y(x),x)^2+1)-c = 0,y(x))
 

\[y \left (x \right ) = \frac {-b +c}{a}\]