29.34.18 problem 1020
Internal
problem
ID
[5589]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1020
Date
solved
:
Sunday, March 30, 2025 at 09:05:20 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2}&=0 \end{align*}
✓ Maple. Time used: 0.073 (sec). Leaf size: 210
ode:=diff(y(x),x)^3+f(x)*(y(x)-a)^2*(y(x)-b)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {1}{\left (\left (-b +\textit {\_a} \right ) \left (\textit {\_a} -a \right )\right )^{{2}/{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{2} \left (y-a \right )^{2}\right )^{{1}/{3}}d \textit {\_a}}{\left (\left (y-b \right ) \left (y-a \right )\right )^{{2}/{3}}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\left (-b +\textit {\_a} \right ) \left (\textit {\_a} -a \right )\right )^{{2}/{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{2} \left (y-a \right )^{2}\right )^{{1}/{3}}d \textit {\_a}}{2 \left (\left (y-b \right ) \left (y-a \right )\right )^{{2}/{3}}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\left (-b +\textit {\_a} \right ) \left (\textit {\_a} -a \right )\right )^{{2}/{3}}}d \textit {\_a} -\frac {\left (i \sqrt {3}-1\right ) \int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-b \right )^{2} \left (y-a \right )^{2}\right )^{{1}/{3}}d \textit {\_a}}{2 \left (\left (y-b \right ) \left (y-a \right )\right )^{{2}/{3}}}+c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.817 (sec). Leaf size: 287
ode=(D[y[x],x])^3 +f[x] (y[x]-a)^2 (y[x]-b)^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-\sqrt [3]{f(K[1])}dK[1]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [-\frac {3 \sqrt [3]{a-\text {$\#$1}} \left (\frac {\text {$\#$1}-b}{a-b}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {a-\text {$\#$1}}{a-b}\right )}{(b-\text {$\#$1})^{2/3}}\&\right ]\left [\int _1^x-(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ] \\
y(x)\to a \\
y(x)\to b \\
\end{align*}
✓ Sympy. Time used: 27.874 (sec). Leaf size: 230
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
f = Function("f")
ode = Eq((-a + y(x))**2*(-b + y(x))**2*f(x) + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ - \frac {\sqrt [3]{-1} \sqrt [3]{- a + y{\left (x \right )}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {\left (- a + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (- a + b \right )}}} \right )}}{\Gamma \left (\frac {4}{3}\right ) \operatorname {polar\_lift}^{\frac {2}{3}}{\left (- a + b \right )}} = C_{1} + \int \sqrt [3]{- f{\left (x \right )}}\, dx, \ - \frac {\sqrt [3]{-1} \sqrt [3]{- a + y{\left (x \right )}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {\left (- a + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (- a + b \right )}}} \right )}}{\Gamma \left (\frac {4}{3}\right ) \operatorname {polar\_lift}^{\frac {2}{3}}{\left (- a + b \right )}} = C_{1} + \frac {i \left (\sqrt {3} + i\right ) \int \sqrt [3]{- f{\left (x \right )}}\, dx}{2}, \ - \frac {\sqrt [3]{-1} \sqrt [3]{- a + y{\left (x \right )}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {2}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {\left (- a + y{\left (x \right )}\right ) e^{2 i \pi }}{\operatorname {polar\_lift}{\left (- a + b \right )}}} \right )}}{\Gamma \left (\frac {4}{3}\right ) \operatorname {polar\_lift}^{\frac {2}{3}}{\left (- a + b \right )}} = C_{1} - \frac {i \left (\sqrt {3} - i\right ) \int \sqrt [3]{- f{\left (x \right )}}\, dx}{2}\right ]
\]