60.2.195 problem 771
Internal
problem
ID
[10769]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
771
Date
solved
:
Sunday, March 30, 2025 at 06:36:09 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {-4 y a x -a^{2} x^{3}-2 a \,x^{2} b -4 a x +8}{8 y+2 a \,x^{2}+4 b x +8} \end{align*}
✓ Maple. Time used: 0.113 (sec). Leaf size: 82
ode:=diff(y(x),x) = (-4*y(x)*a*x-a^2*x^3-2*a*x^2*b-4*a*x+8)/(8*y(x)+2*a*x^2+4*b*x+8);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-a \,x^{2} b -2 b^{2} x +4 \,{\mathrm e}^{\frac {-4 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {b^{2} x}{4}-\frac {b^{2} c_1}{2 a}-\frac {b}{2}-1}}{2}\right ) a +\left (-b^{2} x -2 b -4\right ) a -2 c_1 \,b^{2}}{4 a}}-4 b -8}{4 b}
\]
✓ Mathematica. Time used: 2.577 (sec). Leaf size: 76
ode=D[y[x],x] == (8 - 4*a*x - 2*a*b*x^2 - a^2*x^3 - 4*a*x*y[x])/(8 + 4*b*x + 2*a*x^2 + 8*y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {a b x^2+8 W\left (-e^{-\frac {b^2 x}{4}-1+c_1}\right )+2 b^2 x+4 b+8}{4 b} \\
y(x)\to -\frac {a b x^2+2 b^2 x+4 b+8}{4 b} \\
\end{align*}
✓ Sympy. Time used: 46.746 (sec). Leaf size: 450
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (-a**2*x**3 - 2*a*b*x**2 - 4*a*x*y(x) - 4*a*x + 8)/(2*a*x**2 + 4*b*x + 8*y(x) + 8),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (- \frac {\sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{2 e}\right ) - 2}{b}, \ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (\frac {\sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{2 e}\right ) - 2}{b}, \ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (- \frac {i \sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{2 e}\right ) - 2}{b}, \ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (\frac {i \sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{2 e}\right ) - 2}{b}, \ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (\frac {\sqrt {2} \left (-1 - i\right ) \sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{4 e}\right ) - 2}{b}, \ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (\frac {\sqrt {2} \left (-1 + i\right ) \sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{4 e}\right ) - 2}{b}, \ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (\frac {\sqrt {2} \left (1 - i\right ) \sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{4 e}\right ) - 2}{b}, \ y{\left (x \right )} = \frac {- \frac {a b x^{2}}{4} - \frac {b^{2} x}{2} - b - 2 W\left (\frac {\sqrt {2} \left (1 + i\right ) \sqrt [8]{e^{- 2 b \left (C_{1} b + b x + 2\right )}}}{4 e}\right ) - 2}{b}\right ]
\]