If d/dt=(tF(t))=1+f(t), what is f'(t)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to solve the differential equation `d(tF(t))/(dt) = 1+f(t).`

You should multiply by dt both sides:

`d(tF(t)) = (1 + f(t))dt`

You should write `d(tF(t)) = (tF(t))'.`

Integrating both sides yields:

`int (tF(t))' = int (1 + f(t))dt =gt (tF(t))= t + int f(t)dt`

Dividing by t both sides yields: `F(t) = 1 + (int f(t)dt)/t =gt t(F(t) - 1) = int f(t) dt`

You need to differentiate both sides with respect to t, hence:

`f(t) = d(t(F(t) - 1))/(dt)`

Differentiate using the product property such that:

`f(t) = F(t) - 1 + t(F(t) - 1)' =gt f(t) = F(t) - 1 + t*F(t)`

`f(t) = F(t)(t+1) - 1`

Hence, under the given conditions, differentiating the function f(t) yields `f(t) = F(t)(t+1) - 1` .

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