In ODEs how use the dx/dt equation and x(t)=x to find solutions?

Question

So if I have

dx/dt=x^2/3    x ∈ real numbers     and   x(0)=0

How do I get to
x(t)=0 when t≤a or

x(t)=(t-a)³ when t>a

as a solution

 

I'm confused about what the notation x(t)=? actually means in relation to the x dot equation

Answer 1

dx/dt=x^2/3

 

x-^2/3 dx = dt

 

3  * x^1/3 = t + c

 

For x(0) = 0

3 * 0 = 0 + c => c = 0

 

Then 3 * x^1/3 = t and x(t) = (t/3)³

x(t) represents the function for x when it depends on the variable t.

 

I am not sure what you mean by "x dot", unless you are referring to dx/dt.

dx/dt = d(x(t))/dt

 

Check

 

dx/dt = 3 * (t/3)² * (1/3) = (t/3)²

                                      = (3*x^1/3/3)² = x^2/3