differential equation - Separable variable

[talha nadeem]

According to an economic model, the growth rate of ? = ?(?) is proportional to ?(?) with a factor of proportionality of 5. If initially ?(0) = 10

 

[Deleted member 4993]

According to an economic model, the growth rate of ? = ?(?) is proportional to ?(?) with a factor of proportionality of 5. If initially ?(0) = 10

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[Metronome]

Asking partially for my own sake although I think it will also help the OP with their modeling: When one finds the term "growth rate" or "growth rate of $y(t)$" in the wild, can it be assumed to refer to $a(t)$ in the linear ODE $y'(t) = a(t)y(t) + f(t)$? To me "rate of growth" sounds identical to "rate of change," and so in some contexts I've interpreted "growth rate" to mean $y'(t)$ in the above ODE.

 

[Deleted member 4993]

Asking partially for my own sake although I think it will also help the OP with their modeling: When one finds the term "growth rate" or "growth rate of $y(t)$" in the wild, can it be assumed to refer to $a(t)$ in the linear ODE $y'(t) = a(t)y(t) + f(t)$? To me "rate of growth" sounds identical to "rate of change," and so in some contexts I've interpreted "growth rate" to mean $y'(t)$ in the above ODE.

Yes - in most cases " growth" is equivalent to "change"

 

[JeffM]

$$5y(t) = y’(t) \implies 5y = \dfrac{dy}{dt} \implies \dfrac{dy}{y} = 5dt \implies WHAT?$$
And as Subhotosh Khan asks, what is your question? Are you trying to find y(t)?