60.2.181 problem 757

Internal problem ID [10755]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 757
Date solved : Sunday, March 30, 2025 at 06:35:05 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 x^{2}-4 x -8}{-8 y+2 x^{2}+4 x -8} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 26

ode:=diff(y(x),x) = (-4*x*y(x)+x^3+2*x^2-4*x-8)/(-8*y(x)+2*x^2+4*x-8); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{2}}{4}+2 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{-\frac {1}{2}-\frac {x}{4}}}{2}\right )+\frac {x}{2}+1 \]

✓ Mathematica. Time used: 0.893 (sec). Leaf size: 49

ode=D[y[x],x] == (-8 - 4*x + 2*x^2 + x^3 - 4*x*y[x])/(-8 + 4*x + 2*x^2 - 8*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{4} \left (8 W\left (-e^{-\frac {x}{4}-1+c_1}\right )+x^2+2 x+4\right ) \\ y(x)\to \frac {1}{4} \left (x^2+2 x+4\right ) \\ \end{align*}

✓ Sympy. Time used: 3.609 (sec). Leaf size: 129

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**3 + 2*x**2 - 4*x*y(x) - 4*x - 8)/(2*x**2 + 4*x - 8*y(x) - 8),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {x^{2}}{4} + \frac {x}{2} + 2 W\left (- \frac {\sqrt [4]{C_{1} e^{- x}}}{2 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = \frac {x^{2}}{4} + \frac {x}{2} + 2 W\left (\frac {\sqrt [4]{C_{1} e^{- x}}}{2 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = \frac {x^{2}}{4} + \frac {x}{2} + 2 W\left (- \frac {i \sqrt [4]{C_{1} e^{- x}}}{2 e^{\frac {1}{2}}}\right ) + 1, \ y{\left (x \right )} = \frac {x^{2}}{4} + \frac {x}{2} + 2 W\left (\frac {i \sqrt [4]{C_{1} e^{- x}}}{2 e^{\frac {1}{2}}}\right ) + 1\right ] \]