29.36.31 problem 1102
Internal
problem
ID
[5658]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
36
Problem
number
:
1102
Date
solved
:
Monday, January 27, 2025 at 01:01:46 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.170 (sec). Leaf size: 69
dsolve(diff(y(x),x)^6+f(x)*(y(x)-a)^5*(y(x)-b)^4 = 0,y(x), singsol=all)
\[
\int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{{5}/{6}} \left (\textit {\_a} -b \right )^{{2}/{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )^{4} \left (y \left (x \right )-a \right )^{5}\right )^{{1}/{6}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{{5}/{6}} \left (y \left (x \right )-b \right )^{{2}/{3}}}+c_{1} = 0
\]
✓ Solution by Mathematica
Time used: 2.195 (sec). Leaf size: 561
DSolve[(D[y[x],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 {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-\sqrt [6]{f(K[1])}dK[1]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x\sqrt [6]{f(K[2])}dK[2]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-\sqrt [3]{-1} \sqrt [6]{f(K[3])}dK[3]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [6]{f(K[4])}dK[4]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-(-1)^{2/3} \sqrt [6]{f(K[5])}dK[5]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {2}{3},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$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*}