The rate of change of a variable is proportional to that variable. How will the variation of that variable with respect to time?

Question

Question

Answer 1

Concept:

Let the variable is x which is varying with time t.

\(\frac{dx}{dt} \propto x\)

Calculation:

Given:

\(\frac{dx}{dt} \propto x\)

\(⇒ \frac{dx}{dt} =kx\)

\(⇒ \int\frac{dx}{x} =\int kdt\)

\(⇒ ln x =kt + p\)

Where p is a constant.

Now, take antilog,

⇒ x = e^{(kt +p)}

⇒ x = e(kt) e^p

Let c = ep

⇒ x = c e(kt)

So, the variation will be exponential.