How is the equality in this PDE achieved?

[Superposed_Cat]

I've got this far on a pde (second last step) but have no idea how they got this equality(I'm a noob), could someone please explain? I was going to put this under homework but it is not homework and it doesn't really fit the template. Thanks in advance.

 

[arildno]

It is an ordinary diff.eq,
where dT/dt=-k*T, for a "k" with fluffy garments.

 

[HallsofIvy]

You can see that $T(t)= Ae^{(ih/E)t}$ does satisfy the equation by differentiating. As arildno says, you can just think of k= (ih/E) and use the fact that the derivtive of $Ae^{kt}$ is $kAe^{kt}$.

If you are asking "how can we get that solution if we didn't notice that?", you need to integrate to go the other way:
$$\dfrac{dT}{dt}= -\dfrac{ih}{E}T$$
separating variables,
$$\dfrac{dT}{T}= -\dfrac{ih}{E}dt$$
$$\int\dfrac{dT}{T}= -\dfrac{ih}{E}dt$$
$$ln(T)= -\dfrac{ih}{E}t+ C$$

Now, take the exponential of both sides:
$$T= e^{-\frac{ih}{E}t}e^C$$
and we let $A= e^C$.

 

[Superposed_Cat]

Thanksbut I got it a while ago now. sorry for not mentioning.