I have the following differential equation

$$\frac{dS}{dt}=S\left(t\right)^{y}-\gamma K\left(t\right)$$

If I had a linear specification like $aS(t)$ instead of $S\left(t\right)^{y}$, I would multiply both sides by $e^{-at}$ but since I have a concave function, I don't have a clue to solve this differential equation for $S(t)$. $\gamma$ and $y$ are constant parameters.

  • $\begingroup$ What exactly do you mean by $\frac{d \dot S}{dt}$? I guess it means $\frac{d^2S}{dt^2}$, though I'm not completely sure. $\endgroup$ – Robert Lewis Dec 19 '17 at 18:50
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    $\begingroup$ Sorry, it was a typo. I edited. $\endgroup$ – optimal control Dec 19 '17 at 18:55
  • $\begingroup$ Is there, perhaps, another typo and the last term on the left is multiplied by $S(t)$? Because that will be a Bernoulli type equation, which is easy. If the last term is $-\gamma K(t)$, then I have no idea. $\endgroup$ – Anders Beta Dec 19 '17 at 19:04
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    $\begingroup$ This is a special case of the Chini Equation (A generalization of the Riccati and Abel's equations), given by: $$\frac{dy(t)}{dt}=f(t)(y(t))^n+g(t)y(t)+h(t)$$ There is no known general method to solve them, however some specific cases have a straightforward approach to solve them. $\endgroup$ – projectilemotion Dec 19 '17 at 20:12

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