#### 2.571   ODE No. 571

$a x^n f\left (y'(x)\right )+x y'(x)-y(x)=0$ Mathematica : cpu = 0.0874542 (sec), leaf count = 116

$\text {Solve}\left [\left \{y(x)=a f(\text {K\549626}) x^n+\text {K\549626} x,x=\left (n f(\text {K\549626})^{\frac {1}{n}-1} \int _1^{\text {K\549626}}-\frac {f(K[1])^{\frac {n-1}{n}-1}}{a n}dK[1]-f(\text {K\549626})^{\frac {1}{n}-1} \int _1^{\text {K\549626}}-\frac {f(K[1])^{\frac {n-1}{n}-1}}{a n}dK[1]+c_1 f(\text {K\549626})^{\frac {1}{n}-1}\right ){}^{\frac {1}{n-1}}\right \},\{y(x),\text {K\549626}\}\right ]$ Maple : cpu = 0.418 (sec), leaf count = 169

$\left \{ [y \left ( {\it \_T} \right ) =a \left ( \left ( {\frac {1}{af \left ( {\it \_T} \right ) n} \left ( \left ( 1-n \right ) \int \! \left ( f \left ( {\it \_T} \right ) \right ) ^{-{n}^{-1}}\,{\rm d}{\it \_T}+{\it \_C1}\,an \right ) } \right ) ^{ \left ( n-1 \right ) ^{-1}} \left ( f \left ( {\it \_T} \right ) \right ) ^{{\frac {1}{n \left ( n-1 \right ) }}} \right ) ^{n}f \left ( {\it \_T} \right ) + \left ( {\frac {1}{af \left ( {\it \_T} \right ) n} \left ( \left ( 1-n \right ) \int \! \left ( f \left ( {\it \_T} \right ) \right ) ^{-{n}^{-1}}\,{\rm d}{\it \_T}+{\it \_C1}\,an \right ) } \right ) ^{ \left ( n-1 \right ) ^{-1}} \left ( f \left ( {\it \_T} \right ) \right ) ^{{\frac {1}{n \left ( n-1 \right ) }}}{\it \_T},x \left ( {\it \_T} \right ) = \left ( {\frac {1}{af \left ( {\it \_T} \right ) n} \left ( \left ( 1-n \right ) \int \! \left ( f \left ( {\it \_T} \right ) \right ) ^{-{n}^{-1}}\,{\rm d}{\it \_T}+{\it \_C1}\,an \right ) } \right ) ^{ \left ( n-1 \right ) ^{-1}} \left ( f \left ( {\it \_T} \right ) \right ) ^{{\frac {1}{n \left ( n-1 \right ) }}}] \right \}$