29.36.28 problem 1099
Internal
problem
ID
[5655]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
36
Problem
number
:
1099
Date
solved
:
Monday, January 27, 2025 at 01:01:35 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{6}&=\left (y-a \right )^{4} \left (y-b \right )^{3} \end{align*}
✓ Solution by Maple
Time used: 0.075 (sec). Leaf size: 273
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 -\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{{1}/{6}}}d \textit {\_a} -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 )^{{1}/{6}}}d \textit {\_a} \right )+i \left (-c_{1} +x \right ) \sqrt {3}+x -c_{1}}{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 )^{{1}/{6}}}d \textit {\_a} \right )+i \left (-c_{1} +x \right ) \sqrt {3}+c_{1} -x}{i \sqrt {3}-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 )^{{1}/{6}}}d \textit {\_a} \right )+i \left (-c_{1} +x \right ) \sqrt {3}+c_{1} -x}{i \sqrt {3}-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 )^{{1}/{6}}}d \textit {\_a} \right )+i \left (-c_{1} +x \right ) \sqrt {3}+x -c_{1}}{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 )^{{1}/{6}}}d \textit {\_a} -c_{1} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.184 (sec). Leaf size: 489
DSolve[(D[y[x],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*}