60.2.232 problem 808

Internal problem ID [10819]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 808
Date solved : Monday, January 27, 2025 at 10:04:59 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{\prime }&=\frac {\left (1+2 y\right ) \left (y+1\right )}{x \left (-2 y-2+x +2 y x \right )} \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 45

dsolve(diff(y(x),x) = 1/x*(1+2*y(x))*(y(x)+1)/(-2*y(x)-2+x+2*x*y(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.272 (sec). Leaf size: 75

DSolve[D[y[x],x] == ((1 + y[x])*(1 + 2*y[x]))/(x*(-2 + x - 2*y[x] + 2*x*y[x])),y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{-\frac {2^{2/3} (2 y(x) x+x+y(x)+1)}{x+2 (x-1) y(x)-2}}\frac {1}{K[1]^3-\frac {3 K[1]}{2^{2/3}}+1}dK[1]+\frac {1}{9} 2^{2/3} \left (\frac {1}{x}+\log (x)-1\right )=c_1,y(x)\right ] \]