Internal problem ID [9198]
Internal file name [OUTPUT/8134_Monday_June_06_2022_01_52_33_AM_90546823/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 865.
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.
\[ \boxed {y^{\prime }-\left (\frac {\ln \left (-1+y\right ) y}{\left (1-y\right ) \ln \left (x \right ) x}-\frac {\ln \left (-1+y\right )}{\left (1-y\right ) \ln \left (x \right ) x}-f \left (x \right )\right ) \left (1-y\right )=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -f \left (x \right ) \ln \left (x \right ) x y+y^{\prime } \ln \left (x \right ) x +f \left (x \right ) \ln \left (x \right ) x -\ln \left (-1+y\right ) y+\ln \left (-1+y\right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {f \left (x \right ) \ln \left (x \right ) x y-f \left (x \right ) \ln \left (x \right ) x +\ln \left (-1+y\right ) y-\ln \left (-1+y\right )}{x \ln \left (x \right )} \end {array} \]
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, (y-1)*ln(x)]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 19
dsolve(diff(y(x),x) = (1/(1-y(x))/ln(x)/x*ln(-1+y(x))*y(x)-1/(1-y(x))/ln(x)/x*ln(-1+y(x))-f(x))*(1-y(x)),y(x), singsol=all)
\[ y \left (x \right ) = x^{c_{1} +\int \frac {f \left (x \right )}{\ln \left (x \right )}d x}+1 \]
✓ Solution by Mathematica
Time used: 0.356 (sec). Leaf size: 87
DSolve[y'[x] == (1 - y[x])*(-f[x] - Log[-1 + y[x]]/(x*Log[x]*(1 - y[x])) + (Log[-1 + y[x]]*y[x])/(x*Log[x]*(1 - y[x]))),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\left (-\frac {f(K[1])}{\log (K[1])}-\frac {\log (y(x)-1)}{K[1] \log ^2(K[1])}\right )dK[1]+\int _1^{y(x)}\left (\frac {1}{(K[2]-1) \log (x)}-\int _1^x-\frac {1}{K[1] (K[2]-1) \log ^2(K[1])}dK[1]\right )dK[2]=c_1,y(x)\right ] \]