Internal problem ID [9096]
Internal file name [OUTPUT/8031_Monday_June_06_2022_01_20_07_AM_90332965/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 762.
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(x)*y+H(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-x \right ) y}{x \left (x +1\right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime }+\ln \left (y\right ) y x +y^{\prime } x +y \ln \left (y\right )-x y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\ln \left (y\right ) y x -y \ln \left (y\right )+x y}{x^{2}+x} \end {array} \]
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 trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 4`[0, y/x]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 22
dsolve(diff(y(x),x) = -(ln(y(x))*x+ln(y(x))-x)*y(x)/x/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{\frac {x +c_{1}}{x}} \left (x +1\right )^{-\frac {1}{x}} \]
✓ Solution by Mathematica
Time used: 0.331 (sec). Leaf size: 26
DSolve[y'[x] == ((x - Log[y[x]] - x*Log[y[x]])*y[x])/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to (x+1)^{-1/x} e^{1-\frac {c_1}{x}} \]