Will separation of variables work in solving this PDE?

[AxiomOfChoice]

Homework Statement

As part of the solution to a HW problem of mine, I have to solve the PDE

$$p_t = -vk^2 p - k \delta p_k,$$

where $p = p(k,t)$ and $v,\delta$ are known constants.

Homework Equations

I tried to look for a solution of the form $p(k,t) = K(k)T(t)$ and found one, but I'm not sure if I need to sum/integrate over my arbitrary constant.

The Attempt at a Solution

Separation of variables gave me the solution

$$p(k,t) = A_0 e^{-ct} k^{c/\delta} e^{-vk^2/2\delta},$$

where $c$ is the constant one gets from the separation of variables and $A_0$ is a constant of integration. But I tried solving the PDE in Mathematica, and got a different solution:

$$p(k,t) = A_0(t-(\log k)/\delta) e^{-vk^2/2\delta}.$$

Can someone explain why Mathematica's answer differs from mine? Also, do I need to perform an integration/sum in $c$ to get the most general solution? I've plugged both my solution and Mathematica's in, and they both work, so I'm thinking I have to go another step to get the general solution.

 

[SammyS]

...

I tried to look for a solution of the form $p(x,t) = X(x)T(t)$ and found one, but I'm not sure if I need to sum/integrate over my arbitrary constant.

The Attempt at a Solution

Separation of variables gave me the solution

$$p(k,t) = A_0 e^{-ct} k^{c/\delta} e^{-vk^2/2\delta},$$
...

Can someone explain why Mathematica's answer differs from mine? Also, do I need to perform an integration/sum in $c$ to get the most general solution? I've plugged both my solution and Mathematica's in, and they both work, so I'm thinking I have to go another step to get the general solution.

Show us some details about how you arrived at your solution.

One thing I noticed:

$$\log\left(e^{-ct} k^{c/\delta}\right)=-c\left(t-\frac{\log(k)}{\delta}\right)$$

 

[AxiomOfChoice]

I'll be happy to. I have a function to go to right now, but I'll be back in a few hours and will type up how I arrived at my separation of variables solution.

 

[AxiomOfChoice]

Ok. So, on to my separation of variables calculations. Note that I have also corrected a typo in my original post, replacing some $x$'s with $k$'s. Writing $p(k,t) = K(k)T(t)$ and plugging into the PDE, we find (after dividing through on both sides by $KT$)

$$\frac{1}{T} \frac{\partial T}{\partial t} = -v^2 k^2 - k \delta \frac{1}{K} \frac{\partial K}{\partial k}.$$

The LHS is a function of t only; the RHS a function of k only. So we can (I think) set them both equal to a constant, which I call $-c$ (anticipating that it will be negative). The equation for T becomes

$$\frac{d T}{d t} = -c T.$$

This has the obvious solution $T(t) = C_0 e^{-ct}$. The equation for K, however, becomes (after canceling the minus sign)

$$v^2 k^2 + k \delta \frac{1}{K} \frac{d K}{d k} = c.$$

This can be rearranged and rewritten as

$$\frac{dK}{K} = \left( \frac{c}{k \delta} - v^2 k \right) dk.$$

Integrating on both sides, we obtain (after combining the two constants we get into one, C^0_1, on the RHS)

$$\log K = \frac{c}{\delta} \log k - \frac{v^2 k^2}{2\delta} + C^0_1.$$

Hence, after exponentiating both sides, we obtain

$$K(k) = C_1 k^{c/\delta} \exp \left( -\frac{v^2 k^2}{2\delta} \right).$$

Now we put them together:

$$p(k,t) = K(k)T(t) = A_0 e^{-ct} k^{c/\delta} \exp \left( -\frac{v^2 k^2}{2\delta} \right),$$

which is what I have listed above. Does anyone see anything wrong with this?

 

[SammyS]

It looks good to me.