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?


  • $\begingroup$ The equation is first order and separable, thus explicitly solvable. See also the eerily similar question math.stackexchange.com/questions/3134853/… . $\endgroup$ Mar 6, 2019 at 10:26
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    $\begingroup$ @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? $\endgroup$
    – Dylan
    Mar 6, 2019 at 11:27
  • $\begingroup$ 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! $\endgroup$
    – benharmer
    Mar 6, 2019 at 13:47

1 Answer 1


@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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