60.2.124 problem 700

Internal problem ID [10711]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 700
Date solved : Monday, January 27, 2025 at 09:28:58 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 62

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

\begin{align*} y &= \frac {\sqrt {x \left (2 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{-\frac {x -1}{2 x}}}{2}\right ) x +x -1\right )}}{x} \\ y &= -\frac {\sqrt {x \left (2 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{-\frac {x -1}{2 x}}}{2}\right ) x +x -1\right )}}{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.260 (sec). Leaf size: 72

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

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