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.

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

...

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

 

[anuttarasammyak]

$$\ln 10 e^{y\ln 10 }dy = \frac{dx}{x}$$
$$e^{y\ln 10 } = \ln |x |+ C$$
$$y= \frac{\ln(\ln |x| + C)}{\ln 10}$$defined for
$$|x| >e^{-C}$$

x,C < 0 cases are added to your result.

 

[Gavran]

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.

The substitution $y_1=10^y$ simplifies the equation $$ 10^ydy=\frac{dx}{x\ln10} $$ to the equation $dy_1=dx/x$.
The general case can be obtained by replacing $10$ in the substitution $y_1=10^y$ with $b$, where $b\gt0$ and $b\neq1$, while the equation $dy_1=dx/x$ remains unchanged.