Differential Equations General Solution

Hello!

My differential equation is given as following:

$\displaystyle \frac{dy}{dt}=\lambda y$

I am then to find the general solution.

Which is:

$\displaystyle \int\frac{dy}{dt}\,dt=\int \lambda y\,dt \implies y+C_1 = \lambda^2+C_2 \implies \color{red}\boxed{y=\lambda^2+C}$

Is this correct? (Wondering)

Addition, if the$\displaystyle \lambda$ is positive, negative or 0 what would happen in these three cases if you let $\displaystyle t \rightarrow \infty$

Any help would be greatly appreciated!