# How to solve the Logistic equation with a sinusoidal oscillating growth rate

I want to solve the differential equation $$\frac{\mathrm{d}N}{\mathrm{d}t}=(\cos(\alpha t)+1)\cdot rN(c-N) - \mu N.$$ Any hints as to what type of differential equation this is or how to solve it?

Thanks

• The equation is first order and separable, thus explicitly solvable. See also the eerily similar question math.stackexchange.com/questions/3134853/… . Mar 6 '19 at 10:26
• @FritsVeerman Eerily similar, but the parenthesis are different, so this one isn't separable. Although the question you linked is from the same OP, so maybe they made a typo here? Mar 6 '19 at 11:27
• Yeah, I made two posts - one where the whole equation is affected by the oscillation and this one where only the regeneration rate (r) is affected by the oscillation. I was able to solve the other equation easily (as it is separable) but I am not sure how to solve this one! Mar 6 '19 at 13:47

@Dylan is right: the equation, in this form, is not separable. However, if you introduce $$n(t) = \frac{1}{N(t)}$$, then you obtain for $$n(t)$$ $$\frac{\text{d} n}{\text{d} t} = \mu n- r(\cos(\alpha t) + 1)(c n - 1),$$ which is a linear first order ODE. Hence, you can solve this equation explicitly using variation of constants. Hope this helps!