Mathematics Asked by GORO1 on January 23, 2021
So I’m given a dynamic system $dy/dt$ = 0.8t with y$_0$ = -0.8.
So the Theorem in the book I’m studying says $dx/dt$ = kx
With initial value x$_0$ and k being an ordinary constant.
x is just an arbitrary variable, surely I can bring this principle to ‘y’ as well, right?
The solution is then $e^{kt}$$x_0$
The understanding is that x will grow or decay exponentially.
The answer I originally got with this understanding was ${-0.8}$$e^{0.8t}$.
Seems simple enough, right? Except that was not the answer. I went online and found the answer, but I do not understand how or why we come to that conclusion and ditch the theorem.
The answer I found was actually $0.4t^2$+C
C being -0.8.
And the final answer being $0.4t^2$ – 0.8.
Why do we end up doing this instead of the theorem I brought up earlier? And with this being used as a tool obviously being Integration. When do I draw the line between using an Integration or an Exponential?
Can I use Integration instead of Exponentials?
You need to be careful with the variables and functions.
The differential equation you which to solve is given by
$$frac{dy(t)}{dt} = 0.8 y(t)$$
and has solution given by (with $y_0 = y(0) = -0.8$)
$$y(t) = -0.8 e^{0.8t}$$
The solution you propose is, however (ignoring initial conditions)
$$y(t) = 0.4t^2 + C$$
Your solution makes sense if the differential equation is
$$frac{dy(t)}{dt} = 0.8 t$$
You need to be very careful about the difference between the two.
Correct answer by Eevee Trainer on January 23, 2021
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