Internal problem ID [3803]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1099.
ODE order: 1.
ODE degree: 6.
CAS Maple gives this as type [_quadrature]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.159 (sec). Leaf size: 241
dsolve(diff(y(x),x)^6 = (y(x)-a)^4*(y(x)-b)^3,y(x), singsol=all)
\begin{align*} y \relax (x ) = a \\ y \relax (x ) = b \\ x -\left (\int _{}^{y \relax (x )}\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 \\ x -\left (\int _{}^{y \relax (x )}\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 )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\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 )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}\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 )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\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 )-c_{1} = 0 \\ x -\left (\int _{}^{y \relax (x )}-\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 \\ \end{align*}
✓ Solution by Mathematica
Time used: 122.265 (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 {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ][c_1-i x] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ][i x+c_1] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [-\sqrt [6]{-1} x+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\sqrt [6]{-1} x+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [-(-1)^{5/6} x+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \text {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [(-1)^{5/6} x+c_1\right ] \\ y(x)\to a \\ y(x)\to b \\ \end{align*}