36.28 problem 1099

Internal problem ID [4311]
Internal file name [OUTPUT/3804_Sunday_June_05_2022_11_04_45_AM_69719270/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 36
Problem number: 1099.
ODE order: 1.
ODE degree: 6.

The type(s) of ODE detected by this program : "quadrature"

Maple gives the following as the ode type

[_quadrature]

\[ \boxed {{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3}=0} \] Solving the given ode for \(y^{\prime }\) results in \(6\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \tag {1} \\ y^{\prime }&=\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \tag {2} \\ y^{\prime }&=\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \tag {3} \\ y^{\prime }&=-\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \tag {4} \\ y^{\prime }&=\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \tag {5} \\ y^{\prime }&=\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \tag {6} \end {align*}

Now each one of the above ODE is solved.

Solving equation (1)

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} &= x +c_{1} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{1} \] Verified OK.

Solving equation (2)

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{2} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} &= x +c_{2} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{2} \] Verified OK.

Solving equation (3)

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{3} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} &= x +c_{3} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{3} \] Verified OK.

Solving equation (4)

Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{4} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} &= x +c_{4} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}-\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{4} \] Verified OK.

Solving equation (5)

Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{5} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} &= x +c_{5} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{5} \] Verified OK.

Solving equation (6)

Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{6} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} &= x +c_{6} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{6} \] Verified OK.

36.28.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}, y^{\prime }=-\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}, y^{\prime }=\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}, y^{\prime }=\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}, y^{\prime }=\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}, y^{\prime }=\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}=-1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\int \left (-1\right )d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =-x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}=-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\int \left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}=-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\int \left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}=\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\int \left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) \left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}=\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\int \left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =-x +c_{1} , \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =x +c_{1} , \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} , \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} , \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} , \int \frac {y^{\prime }}{\left (y^{7}-4 y^{6} a -3 y^{6} b +6 y^{5} a^{2}+12 y^{5} a b +3 y^{5} b^{2}-4 y^{4} a^{3}-18 y^{4} a^{2} b -12 y^{4} a \,b^{2}-y^{4} b^{3}+y^{3} a^{4}+12 y^{3} a^{3} b +18 y^{3} a^{2} b^{2}+4 y^{3} a \,b^{3}-3 y^{2} a^{4} b -12 y^{2} a^{3} b^{2}-6 y^{2} a^{2} b^{3}+3 y a^{4} b^{2}+4 y a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d x =\left (\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right ) x +c_{1} \right \} \end {array} \]

Maple trace

`Methods for first order ODEs: 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   -> Solving 1st order ODE of high degree, 1st attempt 
   trying 1st order WeierstrassP solution for high degree ODE 
   trying 1st order WeierstrassPPrime solution for high degree ODE 
   trying 1st order JacobiSN solution for high degree ODE 
   trying 1st order ODE linearizable_by_differentiation 
   trying differential order: 1; missing variables 
   <- differential order: 1; missing  x  successful`
 

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 281

dsolve(diff(y(x),x)^6 = (y(x)-a)^4*(y(x)-b)^3,y(x), singsol=all)
 

\begin{align*} 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 )-c_{1} &= 0 \\ \frac {2 \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 )+i \left (x -c_{1} \right ) \sqrt {3}-c_{1} +x}{1+i \sqrt {3}} &= 0 \\ \frac {-2 \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 )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{-1+i \sqrt {3}} &= 0 \\ \frac {2 \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 )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{-1+i \sqrt {3}} &= 0 \\ \frac {-2 \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 )+i \left (x -c_{1} \right ) \sqrt {3}-c_{1} +x}{1+i \sqrt {3}} &= 0 \\ x +\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} -c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.108 (sec). Leaf size: 489

DSolve[(y'[x])^6 == (y[x]-a)^4 (y[x]-b)^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ][c_1-i x] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ][i x+c_1] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [-\sqrt [6]{-1} x+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\sqrt [6]{-1} x+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [-(-1)^{5/6} x+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [(-1)^{5/6} x+c_1\right ] \\ y(x)\to a \\ y(x)\to b \\ \end{align*}