#### 2.85   ODE No. 85

$y'(x)-x^{a-1} y(x)^{1-b} f\left (\frac {x^a}{a}+\frac {y(x)^b}{b}\right )=0$ Mathematica : cpu = 0.340444 (sec), leaf count = 238

$\text {Solve}\left [\int _1^{y(x)}\left (-\frac {K[2]^{b-1}}{f\left (\frac {x^a}{a}+\frac {K[2]^b}{b}\right )+1}-\int _1^x\left (\frac {K[1]^{a-1} K[2]^{b-1} f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1}-\frac {f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right ) K[1]^{a-1} K[2]^{b-1} f'\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )}{\left (f\left (\frac {K[1]^a}{a}+\frac {K[2]^b}{b}\right )+1\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right ) K[1]^{a-1}}{f\left (\frac {K[1]^a}{a}+\frac {y(x)^b}{b}\right )+1}dK[1]=c_1,y(x)\right ]$ Maple : cpu = 0.348 (sec), leaf count = 153

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