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Solve the following differential equation?

Posted 5 months ago
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I am trying to solve this differential equation

r'[t]=k(1-r[t]/25] . r[0]=20 Where t is time in months from 0 to 12.

the problem is that the coefficient k is also time-dependent. It has values k (Jan, Feb, Mar) = 0.1 k (Apr, May, Jun) = 0.3 k (Jul, Aug, Sep) = 0.5 k (Oct, Nov, Dec) = 0.2

Many thanks in advance!

3 Replies
k[t_] := Piecewise[{{.1, 0 < t < 3}, {.3, 3 < t < 6}, {0.5, 
    6 < t < 9}}, 0.2]
sol = NDSolve[{r'[t] == k[t]*(1 - r[t]/25), r[0] == 20}, 
  r, {t, 0, 12}]
Plot[Evaluate[r[t] /. sol], {t, 0, 12}]

fig1

It can also be solved symbolically just by replacing NDSolve with DSolve (at least in V10+):

DSolve[{r'[t] == k[t]*(1 - r[t]/25), r[0] == 20}, r, {t, 0, 12}]

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