Problem involving ordinary differential equation.

[chwala]

Homework Statement

Consider the ode:
which admits the solution which admits the solution

I am interested in 1. Any-non obvious substitution that simplifies the equation further. 2. Possible generalisations to other logarithmic / exponential bases. 3 Asymptotic behavior of solutions.

Relevant Equations

variable separable.

$10^y dy = \dfrac{dx}{x\ln 10}$
$\int 10^y dy = \int \dfrac{dx}{x\ln 10}$

...

$10^y = \ln x +C$
$10^y = \ln (ax)$
$y=\lg(\ln(ax))$