Internal problem ID [8974]
Internal file name [OUTPUT/7909_Monday_June_06_2022_12_53_53_AM_30059271/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 640.
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.
\[ \boxed {y^{\prime }-\frac {y}{\ln \left (\ln \left (y\right )\right )-\ln \left (x \right )+1}=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \ln \left (x \right )-y^{\prime } \ln \left (\ln \left (y\right )\right )+y-y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y}{-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )-1} \end {array} \]
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`[0, y*(ln(y)*ln(x)-ln(y)*ln(ln(y))-ln(y)+x)/(ln(x)-ln(ln(y))-1)]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
dsolve(diff(y(x),x) = 1/(ln(ln(y(x)))-ln(x)+1)*y(x),y(x), singsol=all)
\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\ln \left (x \right )-\ln \left (\ln \left (\textit {\_a} \right )\right )-1}{\left (-\ln \left (\textit {\_a} \right ) \ln \left (\ln \left (\textit {\_a} \right )\right )+\left (-1+\ln \left (x \right )\right ) \ln \left (\textit {\_a} \right )+x \right ) \textit {\_a}}d \textit {\_a} -c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.221 (sec). Leaf size: 53
DSolve[y'[x] == y[x]/(1 - Log[x] + Log[Log[y[x]]]),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {\log (x)-\log (\log (K[1]))-1}{K[1] (x+\log (x) \log (K[1])-\log (K[1])-\log (K[1]) \log (\log (K[1])))}dK[1]=c_1,y(x)\right ] \]