Answer:
[tex]F(x,t) = C = xe^{-\frac{4}{3}t^6}[/tex]
Step-by-step explanation:
We are given the following in the question:
[tex]\dfrac{dx}{dt} = 8xt^5[/tex]
We solve the above differential equation, with the help of separation of variables.
[tex]\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}[/tex]
Solution:
[tex]F(x,t) = C = xe^{-\frac{4}{3}t^6}[/tex]
