23.2.328 problem 356
Internal
problem
ID
[5683]
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
:
356
Date
solved
:
Tuesday, September 30, 2025 at 01:57:11 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 )^{3}&=0 \end{align*}
✓ Maple. Time used: 0.084 (sec). Leaf size: 69
ode:=diff(y(x),x)^6+f(x)*(y(x)-a)^5*(y(x)-b)^3 = 0;
dsolve(ode,y(x), singsol=all);
\[
\int _{}^{y}\frac {1}{\sqrt {\textit {\_a} -b}\, \left (\textit {\_a} -a \right )^{{5}/{6}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{3} \left (y-a \right )^{5}\right )^{{1}/{6}}d \textit {\_a}}{\sqrt {y-b}\, \left (y-a \right )^{{5}/{6}}}+c_1 = 0
\]
✓ Mathematica. Time used: 2.811 (sec). Leaf size: 567
ode=(D[y[x],x])^6 +f[x]* (y[x]-a)^5 *(y[x]-b)^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x-i \sqrt [6]{f(K[1])}dK[1]+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^xi \sqrt [6]{f(K[2])}dK[2]+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x-\sqrt [6]{-1} \sqrt [6]{f(K[3])}dK[3]+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x\sqrt [6]{-1} \sqrt [6]{f(K[4])}dK[4]+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x-(-1)^{5/6} \sqrt [6]{f(K[5])}dK[5]+c_1\right ]\\ y(x)&\to \text {InverseFunction}\left [-\frac {6 \sqrt [6]{a-\text {$\#$1}} \sqrt {\frac {\text {$\#$1}-b}{a-b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {a-\text {$\#$1}}{a-b}\right )}{\sqrt {b-\text {$\#$1}}}\&\right ]\left [\int _1^x(-1)^{5/6} \sqrt [6]{f(K[6])}dK[6]+c_1\right ]\\ y(x)&\to a\\ y(x)&\to b \end{align*}
✓ Sympy. Time used: 29.882 (sec). Leaf size: 377
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
f = Function("f")
ode = Eq((-a + y(x))**5*(-b + y(x))**3*f(x) + Derivative(y(x), x)**6,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\text {Solution too large to show}
\]