60.1.283 problem 289
Internal
problem
ID
[10297]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
289
Date
solved
:
Wednesday, March 05, 2025 at 10:02:27 AM
CAS
classification
:
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\begin{align*} \left (6 y-x \right )^{2} y^{\prime }-6 y^{2}+2 x y+a&=0 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 113
ode:=(6*y(x)-x)^2*diff(y(x),x)-6*y(x)^2+2*x*y(x)+a = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-x^{3}-18 a x -18 c_{1} \right )^{{1}/{3}}}{6}+\frac {x}{6} \\
y &= -\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 &= -\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: 0.689 (sec). Leaf size: 115
ode=(6*y[x]-x)^2*D[y[x],x]-6*y[x]^2+2*x*y[x]+a==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: 3.297 (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)
\[
\left [ y{\left (x \right )} = \frac {x}{6} + \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}}{2}, \ y{\left (x \right )} = \frac {x}{6} + \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}}{2}, \ y{\left (x \right )} = \frac {x}{6} + \sqrt [3]{C_{1} - \frac {a x}{12} - \frac {x^{3}}{216}}\right ]
\]