#### 2.73   ODE No. 73

$y'(x)-\left (\frac {\text {a0}+\text {a1} x+\text {a2} x^2+\text {a3} x^3}{\text {a0}+\text {a1} y(x)+\text {a2} y(x)^2+\text {a3} y(x)^3}\right )^{2/3}=0$ Mathematica : cpu = 1.33889 (sec), leaf count = 733

$\text {Solve}\left [\frac {3 (\text {a0}+y(x) (\text {a1}+y(x) (\text {a2}+\text {a3} y(x))))^{2/3} \left (y(x)-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,1\right ]\right ) F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};\frac {\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-y(x)}{\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]},\frac {\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-y(x)}{\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,3\right ]}\right )}{5 \left (\frac {y(x)-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,2\right ]}{\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,2\right ]}\right )^{2/3} \left (\frac {y(x)-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,3\right ]}{\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,3\right ]}\right )^{2/3}}=\frac {3 (\text {a0}+x (\text {a1}+x (\text {a2}+\text {a3} x)))^{2/3} \left (x-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,1\right ]\right ) F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};\frac {\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-x}{\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,2\right ]},\frac {\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-x}{\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {a3} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {a1} \text {\#1}+\text {a0}\& ,3\right ]}\right )}{5 \left (\frac {x-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,2\right ]}{\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,2\right ]}\right )^{2/3} \left (\frac {x-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,3\right ]}{\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,1\right ]-\text {Root}\left [\text {\#1}^3 \text {a3}+\text {\#1}^2 \text {a2}+\text {\#1} \text {a1}+\text {a0}\& ,3\right ]}\right )^{2/3}}+c_1,y(x)\right ]$ Maple : cpu = 0.246 (sec), leaf count = 91

$\left \{ \int ^{y \left ( x \right ) }\! \left ( {{\it \_a}}^{3}{\it a3}+{{\it \_a}}^{2}{\it a2}+{\it \_a}\,{\it a1}+{\it a0} \right ) ^{{\frac {2}{3}}}{d{\it \_a}}+\int ^{x}\!- \left ( {\frac {{{\it \_a}}^{3}{\it a3}+{{\it \_a}}^{2}{\it a2}+{\it \_a}\,{\it a1}+{\it a0}}{{\it a3}\, \left ( y \left ( x \right ) \right ) ^{3}+{\it a2}\, \left ( y \left ( x \right ) \right ) ^{2}+{\it a1}\,y \left ( x \right ) +{\it a0}}} \right ) ^{{\frac {2}{3}}} \left ( {\it a3}\, \left ( y \left ( x \right ) \right ) ^{3}+{\it a2}\, \left ( y \left ( x \right ) \right ) ^{2}+{\it a1}\,y \left ( x \right ) +{\it a0} \right ) ^{{\frac {2}{3}}}{d{\it \_a}}+{\it \_C1}=0 \right \}$