# Can I use Separation of Variables like this? (3 terms)

1. Jun 27, 2011

### TylerH

$$\frac{\partial z}{\partial t}=\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$$
$$\mbox{Let }z=T(t)X(x)Y(y)$$
$$T'(t)X(x)Y(y)=T(t)X'(x)Y(y)+T(t)X(x)Y'(y) \Rightarrow$$
$$\frac{T'(t)}{T(t)}=\frac{X'(x)}{X(x)}+\frac{Y'(y)}{Y(y)} \Rightarrow$$
$$\frac{T'(t)}{T(t)}=A \wedge \left(\frac{X'(x)}{X(x)}+\frac{Y'(y)}{Y(y)}=B \rightarrow \frac{X'(x)}{X(x)}=C \wedge \frac{Y'(y)}{Y(y)}=D \right)$$

... and so on.

2. Jun 27, 2011

### LCKurtz

Yes, separation of variables like that sometimes works. For example, the 2d wave equation for a vibrating drumhead is solved that way. Remember, when you separate the variables like that it will only work if the PDE happens to have solutions like that. So you can always try. Maybe it will work on a particular problem and maybe it won't.

3. Jun 28, 2011

### TylerH

$$f(x) \frac{\partial z}{\partial x} = g(y) \frac{\partial z}{\partial y}$$
$$z=X(x)Y(y)$$
$$f(x)X'(x)Y(y)=g(y)X(x)Y'(y)$$
$$\frac{f(x)X'(x)}{X(x)}=\frac{g(y)Y'(y)}{Y(y)}$$
$$\frac{f(x)X'(x)}{X(x)}=A \wedge \frac{g(y)Y'(y)}{Y(y)}=B$$
... and so on.

4. Jun 29, 2011

### TylerH

... anyone?

5. Jun 29, 2011

### LCKurtz

In this case the left side can't depend on y and the right on x so they are both constant. If f and g are not zero then zx=C/f(x) and zy=C/g(y).

Integrating the first gives z(x,y) = F(x) + G(y) where F is the antiderivative of C/f(x).
Differentiating this with respect to y gives G'(y) = C/g(y) so your solution is

z(x,y)=F(x)+G(y) +D

Last edited: Jun 29, 2011
6. Jun 29, 2011

### TylerH

OH!!! I was going about it the wrong way, but with the right technique, I see.

Can you, please, give an example of a PDE that can't be solved by separation (or the general form of one)?