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.

 

[blue_raver22]

It is difficult to provide an accurate response without more context or information about the specific PDE being referenced. However, generally speaking, solving a PDE involves applying various mathematical techniques and principles, such as separation of variables, integration, and boundary conditions. It is important to carefully follow the steps and make sure that all calculations are correct in order to reach the desired equality. If you are struggling with a particular step or concept, it may be helpful to consult with a professor or tutor for further explanation and guidance. Additionally, practicing and reviewing similar problems can also help improve understanding and proficiency in solving PDEs.