Internal problem ID [8952]
Internal file name [OUTPUT/7887_Monday_June_06_2022_12_50_40_AM_15124348/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 618.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-\frac {\left (y+1\right ) \left (\left (y-\ln \left (y+1\right )-\ln \left (x \right )\right ) x +1\right )}{y x}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x y y^{\prime }-y^{2} x +y \ln \left (y+1\right ) x +y \ln \left (x \right ) x -x y+x \ln \left (y+1\right )+x \ln \left (x \right )-y-1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y^{2} x -y \ln \left (y+1\right ) x -y \ln \left (x \right ) x +x y-x \ln \left (y+1\right )-x \ln \left (x \right )+y+1}{x y} \end {array} \]
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 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`[1, (y+1)/x/y]
✓ Solution by Maple
Time used: 0.594 (sec). Leaf size: 36
dsolve(diff(y(x),x) = (y(x)+1)*((y(x)-ln(y(x)+1)-ln(x))*x+1)/y(x)/x,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1}}{x}\right )-1 \\ y \left (x \right ) &= -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{c_{1} {\mathrm e}^{x}-1}}{x}\right )-1 \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.182 (sec). Leaf size: 25
DSolve[y'[x] == ((1 + y[x])*(1 + x*(-Log[x] - Log[1 + y[x]] + y[x])))/(x*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -1-W\left (-\frac {e^{-1+c_1 e^x}}{x}\right ) \]