Internal problem ID [9045]
Internal file name [OUTPUT/7980_Monday_June_06_2022_01_07_12_AM_71729067/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 711.
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.
\[ \boxed {y^{\prime }+\frac {\left (\ln \left (y\right ) x +\ln \left (y\right )-1\right ) y}{x +1}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \ln \left (y\right ) y x +y^{\prime } x +\ln \left (y\right ) y+y^{\prime }-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-\ln \left (y\right ) y x -\ln \left (y\right ) y+y}{x +1} \end {array} \]
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 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] <- symmetry pattern of the form [0, F(x)*G(y)] successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 27
dsolve(diff(y(x),x) = -(ln(y(x))*x+ln(y(x))-1)*y(x)/(x+1),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{{\mathrm e}^{-x} c_{1} -\operatorname {expIntegral}_{1}\left (-x -1\right ) {\mathrm e}^{-x -1}} \]
✓ Solution by Mathematica
Time used: 0.501 (sec). Leaf size: 24
DSolve[y'[x] == ((1 - Log[y[x]] - x*Log[y[x]])*y[x])/(1 + x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{e^{-x-1} (\operatorname {ExpIntegralEi}(x+1)+e c_1)} \]