Question

calculate the flow line x(t) of the given
vector field F that passes through the indicated point at the specified value of t.

F(x, y, z) = 2 i − 3y j + z^3k; x(0) = (3, 5, 7)

Plz tell me how to do it.
I want the whole steps.

Answer 1

Let a(t) be the vector function flow line such that
\[a \prime(t)=F(a(t))\]
Assume that
\[a(t)=<x(t), y(t), z(t)> \exists \space for \space somw \space function\]
\[x'(t)=x(t) \longrightarrow x(t)=Ae^{2t}\]
\[y'(t)=-3y(t)=Be^{-3t}\]

Answer 2

i think u can do the rest work for z(t), then apply the initial conditions or IVP of the derived differential equation describing the vector field then find the values of A, B and C, to find the vector field line in terms of <exp...., exp... exp> =a(t)

Answer 3

This is the solution u have no excuse not to solve this simple system of vector calculus