Internal problem ID [9034]
Internal file name [OUTPUT/7969_Monday_June_06_2022_01_02_57_AM_55997150/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 700.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {1}{x \left (y^{2} x +1+x \right ) y}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } y^{3} x^{2}+y^{\prime } y x^{2}+x y y^{\prime }-1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{y^{3} x^{2}+x^{2} y+x y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati <- Bernoulli successful <- inverse_Riccati successful`
✓ Solution by Maple
Time used: 0.032 (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 \left (x \right ) &= \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 \left (x \right ) &= -\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: 60.147 (sec). Leaf size: 72
DSolve[y'[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*}