60.2.198 problem 774
Internal
problem
ID
[10772]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
774
Date
solved
:
Sunday, March 30, 2025 at 06:36:21 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {-4 x y-x^{3}-2 a \,x^{2}-4 x +8}{8 y+2 x^{2}+4 a x +8} \end{align*}
✓ Maple. Time used: 0.111 (sec). Leaf size: 47
ode:=diff(y(x),x) = (-4*x*y(x)-x^3-2*a*x^2-4*x+8)/(8*y(x)+2*x^2+4*a*x+8);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-2 a^{2} x -a \,x^{2}-8 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1+\frac {\left (c_1 -x \right ) a^{2}}{4}-\frac {a}{2}}}{2}\right )-4 a -8}{4 a}
\]
✓ Mathematica. Time used: 1.125 (sec). Leaf size: 72
ode=D[y[x],x] == (8 - 4*x - 2*a*x^2 - x^3 - 4*x*y[x])/(8 + 4*a*x + 2*x^2 + 8*y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {8 W\left (-e^{-\frac {a^2 x}{4}-1+c_1}\right )+2 a^2 x+a \left (x^2+4\right )+8}{4 a} \\
y(x)\to -\frac {2 a^2 x+a \left (x^2+4\right )+8}{4 a} \\
\end{align*}
✓ Sympy. Time used: 46.080 (sec). Leaf size: 437
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (-2*a*x**2 - x**3 - 4*x*y(x) - 4*x + 8)/(4*a*x + 2*x**2 + 8*y(x) + 8),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (- \frac {\sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{2 e}\right ) - 2}{a}, \ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (\frac {\sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{2 e}\right ) - 2}{a}, \ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (- \frac {i \sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{2 e}\right ) - 2}{a}, \ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (\frac {i \sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{2 e}\right ) - 2}{a}, \ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (\frac {\sqrt {2} \left (-1 - i\right ) \sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{4 e}\right ) - 2}{a}, \ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (\frac {\sqrt {2} \left (-1 + i\right ) \sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{4 e}\right ) - 2}{a}, \ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (\frac {\sqrt {2} \left (1 - i\right ) \sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{4 e}\right ) - 2}{a}, \ y{\left (x \right )} = \frac {- \frac {a^{2} x}{2} - \frac {a x^{2}}{4} - a - 2 W\left (\frac {\sqrt {2} \left (1 + i\right ) \sqrt [8]{e^{- 2 a \left (C_{1} a + a x + 2\right )}}}{4 e}\right ) - 2}{a}\right ]
\]