# Thread: Separation of Variables

1. ## Separation of Variables

$u_{xyz}-xyzu=0$

$\varphi'(x)\psi'(y)\omega'(z)-xyz\varphi(x)\psi(y)\omega(z)=0$

$\displaystyle\frac{\varphi'(x)}{\varphi(x)}=\frac{ xyz\psi(y)\omega(z)}{\psi'(y)\omega'(z)}$

$\varphi'(x)-\lambda\varphi(x)=0\Rightarrow m=\lambda$

$\varphi(x)=\exp{(x\lambda)}$

$\lambda\psi'(y)\omega'(z)=xyz\psi(y)\omega(z)$

$\displaystyle\frac{\psi'(y)}{\psi(y)}=\frac{xyz\om ega(z)}{\lambda\omega'(z)}$

$\psi'(y)-\mu\psi(y)=0\Rightarrow n=\mu$

$\psi(y)=\exp{(y\mu)}$

$\displaystyle\int\frac{\omega'(z)}{\omega(z)}=\int \frac{xyz}{\lambda\mu}dz\Rightarrow \ln{|\omega(z)|}=\frac{xyz^2}{2\lambda\mu}+c$

$\displaystyle\omega(z)=C_2\exp{\left(\frac{xyz^2}{ 2\lambda\mu}\right)}$

$\displaystyle u(x,y,z)=\varphi(x)\psi(y)\omega(z)=C\exp{\left(x\ lambda+y\mu+\frac{xyz^2}{2\lambda\mu}\right)}$

However, this solution doesn't check out.

2. Originally Posted by dwsmith
$u_{xyz}-xyzu=0$

$\varphi'(x)\psi'(y)\omega'(z)-xyz\varphi(x)\psi(y)\omega(z)=0$

$\displaystyle\frac{\varphi'(x)}{\varphi(x)}=\frac{ xyz\psi(y)\omega(z)}{\psi'(y)\omega'(z)}$

$\varphi'(x)-\lambda\varphi(x)=0\Rightarrow m=\lambda$

$\varphi(x)=\exp{(x\lambda)}$

$\lambda\psi'(y)\omega'(z)=xyz\psi(y)\omega(z)$

$\displaystyle\frac{\psi'(y)}{\psi(y)}=\frac{xyz\om ega(z)}{\lambda\omega'(z)}$

$\psi'(y)-\mu\psi(y)=0\Rightarrow n=\mu$

$\psi(y)=\exp{(y\mu)}$

$\displaystyle\int\frac{\omega'(z)}{\omega(z)}=\int \frac{xyz}{\lambda\mu}dz\Rightarrow \ln{|\omega(z)|}=\frac{xyz^2}{2\lambda\mu}+c$

$\displaystyle\omega(z)=C_2\exp{\left(\frac{xyz^2}{ 2\lambda\mu}\right)}$

$\displaystyle u(x,y,z)=\varphi(x)\psi(y)\omega(z)=C\exp{\left(x\ lambda+y\mu+\frac{xyz^2}{2\lambda\mu}\right)}$

However, this solution doesn't check out.
There's a mistake in this line

$\displaystyle\frac{\varphi'(x)}{\varphi(x)}=\frac{ xyz\psi(y)\omega(z)}{\psi'(y)\omega'(z)}$

It should be

$\displaystyle\frac{\varphi'(x)}{x\varphi(x)}=\frac {yz\psi(y)\omega(z)}{\psi'(y)\omega'(z)}$

Notice the location of the $x$.

3. So for psi, the y should be with it as well then?

4. Yep you got it! BTW - what book are you going through and are you doing it independently?

5. Originally Posted by Danny
Yep you got it! BTW - what book are you going through and are you doing it independently?
Elementary Partial Differential Equations by Berg and McGregor 1966. Yes, independently.

6. Originally Posted by dwsmith
Elementary Partial Differential Equations by Berg and McGregor 1966. Yes, independently.
Wow - that's an old book. Couldn't you find anything more recent?

7. Originally Posted by Danny
Wow - that's an old book. Couldn't you find anything more recent?
That is the one my prof likes.

8. I don't mind the book but it lacks in examples.