Question

What is a solution to the differential equation `dx/dt=t(x-2)` with x(0)=5?

Answer 1

`x = 2+3e^(1/2t^2)`

Explanation:

`dx/dt =t(x-2)`, so `x=x(t)`

This is a First Order separable DE, so we can "separate the variables" to get;

`int 1/(x-2)dx =int tdt`

We can easily integrate this to get:
`ln(x-2) =1/2t^2 + C`

Using the initial condition `x(0)=5` (or `x=5` when `t=0`) we get;
`ln(5-2) = 0 + C => C = ln3`

So, `ln(x-2) =1/2t^2 + ln3`
`:. ln(x-2) -ln 3 = 1/2t^2`
`:. ln((x-2)/3) = 1/2t^2`
`:. (x-2)/3 = e^(1/2t^2)`
`:. x-2 = 3e^(1/2t^2)`
`:. x = 2+3e^(1/2t^2)`