# How do I integrate an unknown function of a variable e.g. $a(t)$?

How would I integrate equations of the following form:

$$\frac{d a(t)}{dt}=ka(t)$$

where $$k$$ is constant.

I have the feeling that this is quite simple, but I seem to be stuck.

My initial thought was that I could do:

$$a(t) da(t) = k dt \iff \frac{a^2(t)}{2} = kt + c$$

But for some reason I don't think this is right. Should I instead do integration by parts as done in Integrating an unknown function?

• $\frac{da(t)}{a(t)}=k\,dt$ could be better Jun 25, 2020 at 10:27

This is quite an infamous differential equation, one of the basic ones you learn in calculus courses. Use separation of variables. $$\frac{a(t)}{dt} = ka(t) \implies \frac{da(t)}{a(t)}= kdt \implies \int \frac{da(t)}{a(t)}=\int k dt$$ and so on.
The general solution to this ODE is $$a(t) = Be^{kt}$$, $$B \in \Bbb{R}$$. Note that you made an algebraic error. On the LHS, it should be $$\dfrac{da}{a}$$, so upon integration you should get $$\ln|a| = kt +c$$.