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- Is the following PDE separable

$$\frac{1}{T} \frac{\partial T}{\partial t} = k \frac{1}{Q} \frac{\partial Q}{\partial q} - \frac{\partial q}{\partial t} \frac{1}{Q} \frac{\partial Q}{\partial q}$$

where ##\frac{\partial q}{\partial t}## can depend on ##q## and ##t##?

Suppose I have 2 variables q and t (time), where q is some reparameterization of x (position) : ##x \to q = x f(t)##.

Suppose I have a partial differential equation :

$$\frac{\partial u(q,t)}{\partial t} = k \frac{\partial u(q,t)}{\partial q}$$

where k = constant

Then I do a separation of variables ## u(q,t) = Q(q)T(t) ##

The differential equation becomes (after some manipulation):

$$\frac{1}{T} \frac{\partial T}{\partial t} = k \frac{1}{Q} \frac{\partial Q}{\partial q} - \frac{\partial q}{\partial t} \frac{1}{Q} \frac{\partial Q}{\partial q}$$

where I have used the fact that ##\frac{\partial Q}{\partial t} = \frac{\partial q}{\partial t} \frac{\partial Q}{\partial q}##

Now the left hand side is only dependent on ##t##, while the right hand side depends on both ##q## and ##t##. Since both sides still depend on ##t##, can I say that

$$(LHS) = (RHS) = g(t)$$

(LHS = Left hand side, RHS = Right hand side, g(t) is some function of time)

Additional question :

I have seen on a research paper where the author says that for the above equation to be separable, ##\frac{\partial q}{\partial t}## has to be constant so that RHS only depends on ##q##, hence ##(LHS) = (RHS) = constant##.

But since ##q## still depends on time (##q = x f(t)##), doesn't this mean RHS still depends on time and it should be ##(LHS) = (RHS) = g(t)## instead?

Suppose I have a partial differential equation :

$$\frac{\partial u(q,t)}{\partial t} = k \frac{\partial u(q,t)}{\partial q}$$

where k = constant

Then I do a separation of variables ## u(q,t) = Q(q)T(t) ##

The differential equation becomes (after some manipulation):

$$\frac{1}{T} \frac{\partial T}{\partial t} = k \frac{1}{Q} \frac{\partial Q}{\partial q} - \frac{\partial q}{\partial t} \frac{1}{Q} \frac{\partial Q}{\partial q}$$

where I have used the fact that ##\frac{\partial Q}{\partial t} = \frac{\partial q}{\partial t} \frac{\partial Q}{\partial q}##

Now the left hand side is only dependent on ##t##, while the right hand side depends on both ##q## and ##t##. Since both sides still depend on ##t##, can I say that

$$(LHS) = (RHS) = g(t)$$

(LHS = Left hand side, RHS = Right hand side, g(t) is some function of time)

Additional question :

I have seen on a research paper where the author says that for the above equation to be separable, ##\frac{\partial q}{\partial t}## has to be constant so that RHS only depends on ##q##, hence ##(LHS) = (RHS) = constant##.

But since ##q## still depends on time (##q = x f(t)##), doesn't this mean RHS still depends on time and it should be ##(LHS) = (RHS) = g(t)## instead?

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