#### 2.550   ODE No. 550

$-a y(x)^s-b x^{\frac {r s}{r-s}}+y'(x)^r=0$ Mathematica : cpu = 0.718201 (sec), leaf count = 488

$\text {Solve}\left [\int _1^{y(x)}\left (\frac {r}{-r x \left (a K[2]^s+b x^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}+s x \left (a K[2]^s+b x^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}+r K[2]}-\int _1^x\left (\frac {a s K[2]^{s-1} \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}-1}}{r K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-s K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-r K[2]}-\frac {r \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}} \left (-\frac {a s^2 K[1] K[2]^{s-1} \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}-1}}{r}+a s K[1] K[2]^{s-1} \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}-1}-r\right )}{\left (r K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-s K[1] \left (a K[2]^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-r K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x\frac {r \left (a y(x)^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}}{r K[1] \left (a y(x)^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-s K[1] \left (a y(x)^s+b K[1]^{\frac {r s}{r-s}}\right )^{\frac {1}{r}}-r y(x)}dK[1]=c_1,y(x)\right ]$ Maple : cpu = 0.762 (sec), leaf count = 60

$\left \{ \left ( -r+s \right ) \int _{{\it \_b}}^{y \left ( x \right ) }\! \left ( x \left ( r-s \right ) \sqrt [r]{a{{\it \_a}}^{s}+b{x}^{{\frac {rs}{r-s}}}}-r{\it \_a} \right ) ^{-1}\,{\rm d}{\it \_a}+\ln \left ( x \right ) -{\it \_C1}=0 \right \}$