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

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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

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1 Answer

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@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!

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