ODE
\[ y'(x)^6=(y(x)-a)^4 (y(x)-b)^3 \] ODE Classification
[_quadrature]
Book solution method
Change of variable
Mathematica ✓
cpu = 0.999114 (sec), leaf count = 569
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{3},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [3]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1-i x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{3},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [3]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1+i x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{3},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [3]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1-\sqrt [6]{-1} x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{3},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [3]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1+\sqrt [6]{-1} x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{3},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [3]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1-(-1)^{5/6} x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [-\frac {\sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} B_{\frac {a-\text {$\#$1}}{a-b}}\left (\frac {1}{3},\frac {1}{2}\right )}{\sqrt {b-\text {$\#$1}} \sqrt [3]{\frac {a-\text {$\#$1}}{a-b}}}\& \right ]\left [c_1+(-1)^{5/6} x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.413 (sec), leaf count = 250
\[ \left \{ x-\int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt [6]{ \left ( {\it \_a}-a \right ) ^{4} \left ( {\it \_a}-b \right ) ^{3}}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!{\frac {-2\,i}{-\sqrt {3}+i}{\frac {1}{\sqrt [6]{- \left ( -{\it \_a}+a \right ) ^{4} \left ( -{\it \_a}+b \right ) ^{3}}}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!{\frac {-2\,i}{\sqrt {3}+i}{\frac {1}{\sqrt [6]{- \left ( -{\it \_a}+a \right ) ^{4} \left ( -{\it \_a}+b \right ) ^{3}}}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!{\frac {2\,i}{-\sqrt {3}+i}{\frac {1}{\sqrt [6]{- \left ( -{\it \_a}+a \right ) ^{4} \left ( -{\it \_a}+b \right ) ^{3}}}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!{\frac {2\,i}{\sqrt {3}+i}{\frac {1}{\sqrt [6]{- \left ( -{\it \_a}+a \right ) ^{4} \left ( -{\it \_a}+b \right ) ^{3}}}}}{d{\it \_a}}-{\it \_C1}=0,x-\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt [6]{- \left ( -{\it \_a}+a \right ) ^{4} \left ( -{\it \_a}+b \right ) ^{3}}}}{d{\it \_a}}-{\it \_C1}=0,y \left ( x \right ) =a,y \left ( x \right ) =b \right \} \] Mathematica raw input
DSolve[y'[x]^6 == (-a + y[x])^4*(-b + y[x])^3,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][(-I)*x + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][I*x + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][-((-1)^(1/6)*x) + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][(-1)^(1/6)*x + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*Sqrt[b - #1])) & ][-((-1)^(5/6)*x) + C[1]]}, {y[x] -> InverseFunction[-((Beta[(a - #1)/(a - b), 1/3, 1/2]*(a - #1)^(1/3)*Sqrt[(-b + #1)/(a - b)])/(((a - #1)/(a - b))^(1/3)*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),'implicit')
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/(-(-_a+a)^4*(-_a+b)^3)^(1/6)/(-3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(-2*I/(-(-_a+a)^4*(-_a+b)^3)^(1/6)/(3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(2*I/(-(-_a+a)^4*(-_a+b)^3)^(1/6)/(3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(2*I/(-(-_a+a)^4*(-_a+b)^3)^(1/6)/(-3^(1/2)+I),_a = y(x))-_C1 = 0, x-Intat(-1/(-(-_a+a)^4*(-_a+b)^3)^(1/6),_a = y(x))-_C1 = 0