Internal problem ID [3806]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 36
Problem number: 1102.
ODE order: 1.
ODE degree: 6.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)*y+H(x)]]]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{6}+f \relax (x ) \left (y-a \right )^{5} \left (y-b \right )^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.13 (sec). Leaf size: 68
dsolve(diff(y(x),x)^6+f(x)*(y(x)-a)^5*(y(x)-b)^4 = 0,y(x), singsol=all)
\[ \int _{}^{y \relax (x )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {5}{6}} \left (\textit {\_a} -b \right )^{\frac {2}{3}}}d \textit {\_a} +\int _{}^{x}-\frac {\left (f \left (\textit {\_a} \right ) \left (-y \relax (x )+b \right )^{4} \left (-y \relax (x )+a \right )^{5}\right )^{\frac {1}{6}}}{\left (y \relax (x )-a \right )^{\frac {5}{6}} \left (y \relax (x )-b \right )^{\frac {2}{3}}}d \textit {\_a} +c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 125.286 (sec). Leaf size: 561
DSolve[(y'[x])^6 +f[x] (y[x]-a)^5 (y[x]-b)^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-\sqrt [6]{f(K[1])}dK[1]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x\sqrt [6]{f(K[2])}dK[2]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-\sqrt [3]{-1} \sqrt [6]{f(K[3])}dK[3]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [6]{f(K[4])}dK[4]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-(-1)^{2/3} \sqrt [6]{f(K[5])}dK[5]+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \text {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x(-1)^{2/3} \sqrt [6]{f(K[6])}dK[6]+c_1\right ] \\ y(x)\to a \\ y(x)\to b \\ \end{align*}