ODE
\[ y'(x)^6=(y(x)-a)^4 (y(x)-b)^3 \] ODE Classification
[_quadrature]
Book solution method
Change of variable
Mathematica ✓
cpu = 1.22918 (sec), leaf count = 479
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ][c_1-i x]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ][i x+c_1]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [-\sqrt [6]{-1} x+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [\sqrt [6]{-1} x+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [-(-1)^{5/6} x+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {4}{3};\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\& \right ]\left [(-1)^{5/6} x+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 1.285 (sec), leaf count = 245
\[\left [y \left (x \right ) = a, y \left (x \right ) = b, x -\left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {2 i}{\left (i-\sqrt {3}\right ) \left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {2 i}{\left (\sqrt {3}+i\right ) \left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {2 i}{\left (\sqrt {3}+i\right ) \left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}\frac {2 i}{\left (i-\sqrt {3}\right ) \left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-\textit {\_C1} = 0, x -\left (\int _{}^{y \left (x \right )}-\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-\textit {\_C1} = 0\right ]\] Mathematica raw input
DSolve[y'[x]^6 == (-a + y[x])^4*(-b + y[x])^3,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(-3*Hypergeometric2F1[1/3, 1/2, 4/3, (a - #1)/(a - b)]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][(-I)*x + C[1]]}, {y[x] -> InverseFunction[(-3*Hypergeometric2F1[1/3, 1/2, 4/3, (a - #1)/(a - b)]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][I*x + C[1]]}, {y[x] -> InverseFunction[(-3*Hypergeometric2F1[1/3, 1/2, 4/3, (a - #1)/(a - b)]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][-((-1)^(1/6)*x) + C[1]]}, {y[x] -> InverseFunction[(-3*Hypergeometric2F1[1/3, 1/2, 4/3, (a - #1)/(a - b)]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][(-1)^(1/6)*x + C[1]]}, {y[x] -> InverseFunction[(-3*Hypergeometric2F1[1/3, 1/2, 4/3, (a - #1)/(a - b)]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][-((-1)^(5/6)*x) + C[1]]}, {y[x] -> InverseFunction[(-3*Hypergeometric2F1[1/3, 1/2, 4/3, (a - #1)/(a - b)]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/Sqrt[b - #1] & ][(-1)^(5/6)*x + C[1]]}}
Maple raw input
dsolve(diff(y(x),x)^6 = (y(x)-a)^4*(y(x)-b)^3, y(x))
Maple raw output
[y(x) = a, y(x) = b, x-Intat(1/((_a-a)^4*(_a-b)^3)^(1/6),_a = y(x))-_C1 = 0, x-Intat(-2*I/(I-3^(1/2))/((_a-a)^4*(_a-b)^3)^(1/6),_a = y(x))-_C1 = 0, x-Intat(-2*I/(3^(1/2)+I)/((_a-a)^4*(_a-b)^3)^(1/6),_a = y(x))-_C1 = 0, x-Intat(2*I/(3^(1/2)+I)/((_a-a)^4*(_a-b)^3)^(1/6),_a = y(x))-_C1 = 0, x-Intat(2*I/(I-3^(1/2))/((_a-a)^4*(_a-b)^3)^(1/6),_a = y(x))-_C1 = 0, x-Intat(-1/((_a-a)^4*(_a-b)^3)^(1/6),_a = y(x))-_C1 = 0]