Integrating the product of an exponential and a derivative

Question

I have the following problem that I'm unsure how to tackle:

$\frac{dm}{dt} = \frac{dn}{dt} - \lambda m$

I tried using the integrating factor method with IF = $e^{\lambda t}$ so I end up with:

$me^{\lambda t} = \int{\frac{dn}{dt}e^{\lambda t} dt}$

However, I'm not sure how to progress with this integral? Can anyone help, please?

Thanks.

edit: $\lambda = constant$

Answer 1

You can cancel out the $dt’s$. $$\int \frac{dn}{dt} e^{\lambda t} dt = \int e^{\lambda t} dn = ne^{\lambda t} + C $$