Partial derivatives

I've done the first part - that was fine. However I am having trouble with the last part... I can't really see how to solve for f since we're not told what f(x) or g(x) are!

Indeed the coefficients of each term matches with the coefficients of the identity we've proven... not sure how to use that.

Any help is appreciated!

Reply 1

qgujxj39

17

When you solve an ordinary differential equation, you get one or more arbitrary constants in the general solution.

When you solve a partial differential equation, you get one or more arbitrary functions in the general solution.

For example, the general solution of the PDE

\dfrac{\partial u}{\partial x} - \dfrac{\partial u}{\partial y} = 0

is

u = \text{f} (x + y)

, where

\text{f}

is any function. You'd then need some initial/boundary conditions to work out what this arbitrary function is for your specific equation (just like you'd need initial/boundary conditions to work out the arbitrary constants for an ODE). Remember that your question is only asking for the general solution.

Reply 2

ztibor

10

Original post by fkhan100

I've done the first part - that was fine. However I am having trouble with the last part... I can't really see how to solve for f since we're not told what f(x) or g(x) are!

Indeed the coefficients of each term matches with the coefficients of the identity we've proven... not sure how to use that.

Any help is appreciated!

This partial diff. eq. with the x=uv and y=u/v szbstitution matches
with the identity with that
H=1 and K=-y/x
because f.e

\frac{\partial x}{\partial u}\cdot \frac{\partial y}{\partial v}+\frac{\partial x}{\partial v}\cdot \frac{\partial y}{\partial u}=v\cdot (-\frac{u}{v^2})+u\cdot \frac{1}{v}=0

with this your equation is

\frac{\partial^2 g}{\partial u\partial v}=0

(edited 10 years ago)

Reply 3

Ashvind Poonith

1

Any help for the first part please?

Reply 4

I hate maths!

12

Original post by Ashvind Poonith

Any help for the first part please?

Which part of the first part? Have you used the multivariate chain rule to find those expressions for ∂g/∂u and ∂g/∂v? Also you might have better luck starting a new thread where your question will get more attention.

Partial derivatives

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