Unique solution of 1st order autonomous, homogeneous DE

[Pietair]

Hello,

1st order autonomous, homogeneous differential equation have the general form:
$$\dot{x}(t)=ax(t)$$

It can be shown that the unique solution is always:

$$x(t)=e^{at}x(t_{0})$$

I don't get this, could anyone help me?

Thanks!

 

[HallsofIvy]

Have you tried at all? What is the derivative of $e^{at}$. Since $x(t_0)$ is a constant, what is the derivative of $e^{at}x(t_0)$?

 

[Pietair]

Thank you for your answer.

I can work it out when x(t) = x, but this is not the case, is it?

 

[Office_Shredder]

x(t) is supposed to be a function of t. What do you mean by x(t)=x?

 

[Pietair]

How can I calculate the integral of x(t) when I don't know the corresponding function? x(t) can equal (t^2) or (t-3) and so on, right?

 

[HallsofIvy]

Will you please answer my questions? Do you know what the derivative of $e^{at}$ is? Do you know what the derivative of $e^{at}x(t_0)$ is?

 

[HallsofIvy]

How can I calculate the integral of x(t) when I don't know the corresponding function? x(t) can equal (t^2) or (t-3) and so on, right?

You can't and you don't want to.

If $$\frac{dx}{dt}= ax$$ then
$$\frac{dx}{x}= adt$$

Integrate both sides of
$$\int \frac{dx}{x}= \int a dt$$

Note that on the left you have NO "t". The only variable is x.