Solving Partial Differential Equations: Understanding a Key Step

[Google_Spider]

[SOLVED] Whoa! How did they do this ?

Hi,

I am trying to teach myself partial diff equations and I'm stuck with this small step which I do not understand. I hope someone will help me.

Homework Statement
Solve the Partial Differential Equation
$$(x^2-y^2-z^2)p+2xyq=2xz$$ Solution given in the book

Subsidiary equations are

$$\frac{dx}{x^2-y^2-z^2}=\frac{dy}{2xy}=\frac{dz}{2xz}$$

The step which I do not understand is this :

Using multipliers x, y, z. we get:
each fraction=$$\frac{xdx+ydy+zdz}{x(x^2+y^2+z^2)}$$

Rest of the solution I understand but I don't get that how they did this particular step. How did they get this fraction ?

Thanks for your time and effort

 

[tiny-tim]

… ratios …

The step which I do not understand is this :

Using multipliers x, y, z. we get:
each fraction=$$\frac{xdx+ydy+zdz}{x(x^2+y^2+z^2)}$$

Rest of the solution I understand but I don't get that how they did this particular step. How did they get this fraction ?

Hi Google_Spider!

It's easy algebra - but it looks extremely unlikely the first time you see it!

If the ratio X/A = Y/B = a, then you can add any multiples together on top and bottom, and the ratio is the same:

(pX + qY)/(pA + qB) = (paA + qaB)/(pA + qB) = a(pA + qB)/(pA + qB) = a.​

And the same works for X/A = Y/B = Z/C = a, and so on.

In this case, put p = x, q = y, r = z, and see what happens …

 

[Google_Spider]

Thanks tiny-tim!