#### 2.518   ODE No. 518

$y'(x)^3-(y(x)-a)^2 (y(x)-b)^2=0$ Mathematica : cpu = 0.517174 (sec), leaf count = 236

$\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {\#1}} \left (\frac {\text {\#1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {a-\text {\#1}}{a-b}\right )}{(b-\text {\#1})^{2/3}}\& \right ][c_1+x]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {\#1}} \left (\frac {\text {\#1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {a-\text {\#1}}{a-b}\right )}{(b-\text {\#1})^{2/3}}\& \right ]\left [c_1-\sqrt [3]{-1} x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {\#1}} \left (\frac {\text {\#1}-b}{a-b}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {a-\text {\#1}}{a-b}\right )}{(b-\text {\#1})^{2/3}}\& \right ]\left [c_1+(-1)^{2/3} x\right ]\right \}\right \}$ Maple : cpu = 0.156 (sec), leaf count = 126

$\left \{ x-\int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt [3]{ \left ( {\it \_a}-a \right ) ^{2} \left ( {\it \_a}-b \right ) ^{2}}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!2\,{\frac {1}{ \left ( i\sqrt {3}-1 \right ) \sqrt [3]{ \left ( -{\it \_a}+a \right ) ^{2} \left ( -{\it \_a}+b \right ) ^{2}}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!-2\,{\frac {1}{ \left ( i\sqrt {3}+1 \right ) \sqrt [3]{ \left ( -{\it \_a}+a \right ) ^{2} \left ( -{\it \_a}+b \right ) ^{2}}}}{d{\it \_a}}-{\it \_C1}=0,y \left ( x \right ) =a,y \left ( x \right ) =b \right \}$