23.2.326 problem 354
Internal
problem
ID
[5681]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
2.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
SECOND
OR
HIGHER
DEGREE,
page
278
Problem
number
:
354
Date
solved
:
Tuesday, September 30, 2025 at 01:57:00 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{6}&=\left (y-a \right )^{4} \left (y-b \right )^{3} \end{align*}
✓ Maple. Time used: 0.152 (sec). Leaf size: 273
ode:=diff(y(x),x)^6 = (y(x)-a)^4*(y(x)-b)^3;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= a \\
y &= b \\
x -\int _{}^{y}\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 \int _{}^{y}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{{1}/{6}}}d \textit {\_a} +i \left (x -c_1 \right ) \sqrt {3}+x -c_1}{1+i \sqrt {3}} &= 0 \\
\frac {-2 \int _{}^{y}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{{1}/{6}}}d \textit {\_a} +i \left (x -c_1 \right ) \sqrt {3}-x +c_1}{i \sqrt {3}-1} &= 0 \\
\frac {2 \int _{}^{y}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{{1}/{6}}}d \textit {\_a} +i \left (x -c_1 \right ) \sqrt {3}-x +c_1}{i \sqrt {3}-1} &= 0 \\
\frac {-2 \int _{}^{y}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{{1}/{6}}}d \textit {\_a} +i \left (x -c_1 \right ) \sqrt {3}+x -c_1}{1+i \sqrt {3}} &= 0 \\
x +\int _{}^{y}\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*}
✓ Mathematica. Time used: 2.851 (sec). Leaf size: 489
ode=(D[y[x],x])^6 == (y[x]-a)^4 *(y[x]-b)^3;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 13.849 (sec). Leaf size: 382
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-(-a + y(x))**4*(-b + y(x))**3 + Derivative(y(x), x)**6,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\text {Solution too large to show}
\]