Question

The solution of the differential equation ${\displaystyle \frac{dx}{dt}=\frac{x\log x}{t}}$ is ______.

Options

• x = e^ct

• x + e^ct = 0

• x = e^t + t

• xe^ct = 0

Answer 1

The solution of the differential equation ${\displaystyle \frac{dx}{dt}=\frac{x\log x}{t}}$ is x = e^ct.

Explanation:

${\displaystyle \frac{dx}{dt}=\frac{x\log x}{t}}$

∴ ${\displaystyle \frac{dx}{x\log x}\frac{dt}{t}}$

Integrating both sides, we get

${\displaystyle \int \frac{dx}{x\log x}=\int \frac{dt}{t}}$

∴ log (log x) = log (t) + log c

∴ log (log x) = log (tc)

∴ log x = ct

∴ e^ct = x