Internal problem ID [9141]
Internal file name [OUTPUT/8076_Monday_June_06_2022_01_40_30_AM_60605069/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 807.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[NONE]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\frac {1}{-x -\textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}+y^{\prime } x -1=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {1}{\textit {\_F1} \left (y-\ln \left (x \right )\right ) y \,{\mathrm e}^{y}+x} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying homogeneous types: differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying an equivalence to an Abel ODE 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, 1/(f__1(y-ln(x))*y*exp(y)+x)*f__1(y-ln(x))*exp(y)]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 43
dsolve(diff(y(x),x) = -1/(-x-_F1(y(x)-ln(x))*y(x)*exp(y(x))),y(x), singsol=all)
\[ \frac {\ln \left (x \right )^{2}}{2}-y \left (x \right ) \ln \left (x \right )-\left (\int _{}^{y \left (x \right )-\ln \left (x \right )}\frac {f_{1} \left (\textit {\_a} \right ) \textit {\_a} +{\mathrm e}^{-\textit {\_a}}}{f_{1} \left (\textit {\_a} \right )}d \textit {\_a} \right )+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.255 (sec). Leaf size: 57
DSolve[y'[x] == -(-x - E^y[x]*F1[-Log[x] + y[x]]*y[x])^(-1),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\int _1^{y(x)-\log (x)}\frac {\text {F1}(K[1]) K[1]+e^{-K[1]}}{\text {F1}(K[1])}dK[1]-y(x) \log (x)+\frac {\log ^2(x)}{2}=-c_1,y(x)\right ] \]