How do you solve this...?

Question

differential equation:

dx/dt=kx

where k is some arbitrary constant.

i tried this

dx=dt kx

∫dx=∫kx dt

x=k∫x dt

i tried integration by parts twice, that didnt tell me anything.

this requires the fundamental theorem of calculus, i assume. but how do i use it to solve this?

Answer 1

This problem is separable into it's "x" component and "t" component

1/x  dx  = k dt

integrate both side

int(1/x)dx    =  int(k) dt

take it from there

good luck

Answer 2

it's saparable

dx/dt = kx

dx/x = k dt

ln lxl = kt + C

x = e^(kt+C)

x = e^(kt) * e^C

x = C e^(kt) (C is a constant since e^C is also a constant)

Answer 3

Assume that x = a exp (b t), where a and b are constants.  Substitute this into the equation, and you will get a and b in terms of k.  The equation is the standard exponential growth or decay equation, used to describe everything from radioactive decay to the growth of populations.