Solve the equation. StartFraction dx Over dt EndFraction equals8 xt Superscript 5 An implicit solution in the form ​F(t,x)equalsC is nothing

Question

Solve the equation. StartFraction dx Over dt EndFraction equals8 xt Superscript 5 An implicit solution in the form ​F(t,x)equalsC is nothingequals​C, where C is an arbitrary constant. ​(Type an expression using t and x as the​ variables.)

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Mia 3 weeks 2022-01-01T14:57:37+00:00 1 Answer 0 views 0

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    2022-01-01T14:59:36+00:00

    Answer:

    F(x,t) = C = xe^{-\frac{4}{3}t^6}

    Step-by-step explanation:

    We are given the following in the question:

    \dfrac{dx}{dt} = 8xt^5

    We solve the above differential equation, with the help of separation of variables.

    \dfrac{dx}{dt} = 8xt^5\\\\\dfrac{dx}{x} = 8t^5~dt\\\\\text{Integrating both sides}\\\\\displaystyle\int\dfrac{dx}{x} = \int 8t^5~dt\\\\\log(x) = \dfrac{4}{3}t^6 +\log C\\\\\text{where C is constant of integration.}\\\\\log \dfrac{x}{C} =  \dfrac{4}{3}t^6\\\\\dfrac{x}{C} = e^{\dfrac{4}{3}t^6}\\\\C = x e^{-\frac{4}{3}t^6}

    Solution:

    F(x,t) = C = xe^{-\frac{4}{3}t^6}

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45:7+7-4:2-5:5*4+35:2 =? ( )