29.34.17 problem 1019
Internal
problem
ID
[5588]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1019
Date
solved
:
Sunday, March 30, 2025 at 09:05:08 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}&=\left (y-a \right )^{2} \left (y-b \right )^{2} \end{align*}
✓ Maple. Time used: 0.133 (sec). Leaf size: 142
ode:=diff(y(x),x)^3 = (y(x)-a)^2*(y(x)-b)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= a \\
y &= b \\
x -\int _{}^{y}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{3}}}d \textit {\_a} -c_1 &= 0 \\
\frac {2 \int _{}^{y}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{3}}}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 )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{{1}/{3}}}d \textit {\_a} +i \left (x -c_1 \right ) \sqrt {3}-x +c_1}{i \sqrt {3}-1} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.12 (sec). Leaf size: 246
ode=(D[y[x],x])^3 ==(y[x]-a)^2 (y[x]-b)^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ][x+c_1] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [-\sqrt [3]{-1} x+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [(-1)^{2/3} x+c_1\right ] \\
y(x)\to a \\
y(x)\to b \\
\end{align*}
✓ Sympy. Time used: 28.418 (sec). Leaf size: 204
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-(-a + y(x))**2*(-b + y(x))**2 + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - \frac {\sqrt [3]{-1} \sqrt [3]{- a + y{\left (x \right )}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {\left (- a + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (- a + b \right )}}} \right )}}{\Gamma \left (\frac {4}{3}\right ) \operatorname {polar\_lift}^{\frac {2}{3}}{\left (- a + b \right )}} = C_{1} + x, \ - \frac {\sqrt [3]{-1} \sqrt [3]{- a + y{\left (x \right )}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {\left (- a + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (- a + b \right )}}} \right )}}{\Gamma \left (\frac {4}{3}\right ) \operatorname {polar\_lift}^{\frac {2}{3}}{\left (- a + b \right )}} = C_{1} + \frac {i x \left (\sqrt {3} + i\right )}{2}, \ - \frac {\sqrt [3]{-1} \sqrt [3]{- a + y{\left (x \right )}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {\left (- a + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (- a + b \right )}}} \right )}}{\Gamma \left (\frac {4}{3}\right ) \operatorname {polar\_lift}^{\frac {2}{3}}{\left (- a + b \right )}} = C_{1} - \frac {i x \left (\sqrt {3} - i\right )}{2}\right ]
\]