29.34.19 problem 1021
Internal
problem
ID
[5590]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1021
Date
solved
:
Sunday, March 30, 2025 at 09:05:23 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} \left (y-c \right )^{2}&=0 \end{align*}
✓ Maple. Time used: 0.184 (sec). Leaf size: 267
ode:=diff(y(x),x)^3+f(x)*(y(x)-a)^2*(y(x)-b)^2*(y(x)-c)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {1}{\left (\left (-c +\textit {\_a} \right ) \left (-b +\textit {\_a} \right ) \left (-a +\textit {\_a} \right )\right )^{{2}/{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y-c \right )^{2} \left (y-b \right )^{2} \left (y-a \right )^{2}\right )^{{1}/{3}}d \textit {\_a}}{\left (\left (y-c \right ) \left (y-b \right ) \left (y-a \right )\right )^{{2}/{3}}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\left (-c +\textit {\_a} \right ) \left (-b +\textit {\_a} \right ) \left (-a +\textit {\_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-c \right )^{2} \left (y-b \right )^{2} \left (y-a \right )^{2}\right )^{{1}/{3}}d \textit {\_a}}{2 \left (\left (y-c \right ) \left (y-b \right ) \left (y-a \right )\right )^{{2}/{3}}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\left (-c +\textit {\_a} \right ) \left (-b +\textit {\_a} \right ) \left (-a +\textit {\_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-c \right )^{2} \left (y-b \right )^{2} \left (y-a \right )^{2}\right )^{{1}/{3}}d \textit {\_a}}{2 \left (\left (y-c \right ) \left (y-b \right ) \left (y-a \right )\right )^{{2}/{3}}}+c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 14.44 (sec). Leaf size: 421
ode=(D[y[x],x])^3 +f[x](y[x]-a)^2 (y[x]-b)^2 (y[x]-c)^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 {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\&\right ]\left [\int _1^x-\sqrt [3]{f(K[1])}dK[1]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\&\right ]\left [\int _1^x\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {3 \sqrt [3]{a-\text {$\#$1}} \sqrt [3]{c-\text {$\#$1}} \left (\frac {(b-\text {$\#$1}) (a-c)}{(c-\text {$\#$1}) (a-b)}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {(c-b) (a-\text {$\#$1})}{(a-b) (c-\text {$\#$1})}\right )}{(b-\text {$\#$1})^{2/3} (a-c)}\&\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 \\
y(x)\to c \\
\end{align*}
✓ Sympy. Time used: 28.867 (sec). Leaf size: 128
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
f = Function("f")
ode = Eq((-a + y(x))**2*(-b + y(x))**2*(-c + y(x))**2*f(x) + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\left (- y + a\right )^{\frac {2}{3}} \left (- y + b\right )^{\frac {2}{3}} \left (- y + c\right )^{\frac {2}{3}}}\, dy = C_{1} + \int \sqrt [3]{- f{\left (x \right )}}\, dx, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\left (- y + a\right )^{\frac {2}{3}} \left (- y + b\right )^{\frac {2}{3}} \left (- y + c\right )^{\frac {2}{3}}}\, dy = C_{1} + \frac {i \left (\sqrt {3} + i\right ) \int \sqrt [3]{- f{\left (x \right )}}\, dx}{2}, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\left (- y + a\right )^{\frac {2}{3}} \left (- y + b\right )^{\frac {2}{3}} \left (- y + c\right )^{\frac {2}{3}}}\, dy = C_{1} - \frac {i \left (\sqrt {3} - i\right ) \int \sqrt [3]{- f{\left (x \right )}}\, dx}{2}\right ]
\]