29.23.2 problem 632
Internal
problem
ID
[5224]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
23
Problem
number
:
632
Date
solved
:
Tuesday, March 04, 2025 at 08:34:28 PM
CAS
classification
:
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\begin{align*} \left (x -6 y\right )^{2} y^{\prime }+a +2 x y-6 y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 113
ode:=(x-6*y(x))^2*diff(y(x),x)+a+2*x*y(x)-6*y(x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= \frac {\left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{6}+\frac {x}{6} \\
y \left (x \right ) &= -\frac {\left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}-\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}+\frac {x}{6} \\
y \left (x \right ) &= -\frac {\left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}+\frac {i \sqrt {3}\, \left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{12}+\frac {x}{6} \\
\end{align*}
✓ Mathematica. Time used: 0.669 (sec). Leaf size: 115
ode=(x-6 y[x])^2 D[y[x],x]+a+2 x y[x]-6 y[x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {1}{6} \left (x+\sqrt [3]{-18 a x-x^3+18 c_1}\right ) \\
y(x)\to \frac {x}{6}+\frac {1}{12} i \left (\sqrt {3}+i\right ) \sqrt [3]{-18 a x-x^3+18 c_1} \\
y(x)\to \frac {x}{6}-\frac {1}{12} \left (1+i \sqrt {3}\right ) \sqrt [3]{-18 a x-x^3+18 c_1} \\
\end{align*}
✓ Sympy. Time used: 3.366 (sec). Leaf size: 83
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a + 2*x*y(x) + (x - 6*y(x))**2*Derivative(y(x), x) - 6*y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {x}{6} + \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}}{2}, \ y{\left (x \right )} = \frac {x}{6} + \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}}{2}, \ y{\left (x \right )} = \frac {x}{6} + \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}\right ]
\]