Internal problem ID [8972]
Internal file name [OUTPUT/7907_Monday_June_06_2022_12_53_28_AM_80074097/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 638.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`x=_G(y,y')`]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }+\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y=0} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\left (-\ln \left (\ln \left (y\right )\right )+\ln \left (x \right )\right ) y \end {array} \]
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 `, `-> Computing symmetries using: way = 5`[0, x*y*ln(x)+ln(y)*y-x*y*ln(ln(y))]
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 36
dsolve(diff(y(x),x) = -(-ln(ln(y(x)))+ln(x))*y(x),y(x), singsol=all)
\[ -\left (\int _{\textit {\_b}}^{y \left (x \right )}-\frac {1}{\textit {\_a} \left (x \ln \left (x \right )-\ln \left (\ln \left (\textit {\_a} \right )\right ) x +\ln \left (\textit {\_a} \right )\right )}d \textit {\_a} \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.109 (sec). Leaf size: 41
DSolve[y'[x] == (-Log[x] + Log[Log[y[x]]])*y[x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{K[1] (x \log (x)+\log (K[1])-x \log (\log (K[1])))}dK[1]=-\log (x)+c_1,y(x)\right ] \]