Internal problem ID [9261]
Internal file name [OUTPUT/8197_Monday_June_06_2022_02_15_08_AM_36407465/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 928.
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.
\[ \boxed {y^{\prime }-\frac {\left ({\mathrm e}^{-\frac {y}{x}} y x +{\mathrm e}^{-\frac {y}{x}} y+{\mathrm e}^{-\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} x +x \right ) {\mathrm e}^{\frac {y}{x}}}{x \left (1+x \right )}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {y}{x}} y \,{\mathrm e}^{\frac {y}{x}} x +{\mathrm e}^{-\frac {y}{x}} {\mathrm e}^{\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} y \,{\mathrm e}^{\frac {y}{x}}+{\mathrm e}^{-\frac {y}{x}} {\mathrm e}^{\frac {y}{x}} x -x^{2} y^{\prime }+x \,{\mathrm e}^{\frac {y}{x}}-y^{\prime } x =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {{\mathrm e}^{-\frac {y}{x}} y \,{\mathrm e}^{\frac {y}{x}} x +{\mathrm e}^{-\frac {y}{x}} {\mathrm e}^{\frac {y}{x}} x^{2}+{\mathrm e}^{-\frac {y}{x}} y \,{\mathrm e}^{\frac {y}{x}}+{\mathrm e}^{-\frac {y}{x}} {\mathrm e}^{\frac {y}{x}} x +x \,{\mathrm e}^{\frac {y}{x}}}{-x^{2}-x} \end {array} \]
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 `, `-> Computing symmetries using: way = 5`[0, exp(y/x)]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 20
dsolve(diff(y(x),x) = (exp(-y(x)/x)*y(x)*x+exp(-y(x)/x)*y(x)+exp(-y(x)/x)*x^2+exp(-y(x)/x)*x+x)*exp(y(x)/x)/x/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = -\ln \left (\frac {-\ln \left (x +1\right )+c_{1}}{x}\right ) x \]
✓ Solution by Mathematica
Time used: 2.195 (sec). Leaf size: 22
DSolve[y'[x] == (E^(y[x]/x)*(x + x/E^(y[x]/x) + x^2/E^(y[x]/x) + y[x]/E^(y[x]/x) + (x*y[x])/E^(y[x]/x)))/(x*(1 + x)),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -x \log \left (\frac {-\log (x+1)+c_1}{x}\right ) \]