# Find the solution of this differential equation

## Homework Statement

if
$$f \left( x,y \right) =xy*g \left( 7\,{x}^{2}-9\,{y}^{2} \right)$$
then the differential equation $$9y\frac{{\partial f}}{{\partial x}} + 7x\frac{{\partial f}}{{\partial y}} = ?$$ is equal to?

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## The Attempt at a Solution

i triedto do a partial derivatives wich doesnt really seem to work
if you can just give me a direction in which to go to
thanks

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tiny-tim
Homework Helper
i triedto do a partial derivatives which doesnt really seem to work
Hi supercali!

I don't see what doesn't really seem to work … which might be because you haven't told us!

(you do know they don't want you to solve the equation, only to write it out?)

i dont think i really understand what it is i need to do

tiny-tim
Homework Helper
Just write out ∂f/∂x, and multiply it by 9y …

(they haven't told you what g' is, so just leave it as g')

(you do understand that g is an ordinary function, don't you?)

no thats not what they want me to do
they want me to form a differential equation the answer suppose to be with f ,x and y
but i just dont think i understand how to do so

Find
1. df/dx
df/dy

g(mess) is a function, so use chain rule for it

2. Substitute df/dx and df/dy in your target equation

I think you misunderstood tiny-tim. You don't write out but do substitution.

You don't need anything else (I can say for sure :) )

tiny-tim
Homework Helper
Hi supercali!

they want me to form a differential equation the answer suppose to be with f ,x and y
No … they want you to form a differential equation with g x and y.

f should disappear

now i have the question infront of me the say there that they want the answer in terms of y x and f not g

tiny-tim
Homework Helper
now i have the question infront of me the say there that they want the answer in terms of y x and f not g
no wonder you're confused …

once you've done ∂/∂x to f, the f has gone!

all that's left is x and y and g and g'

Anyway, as rootX says, just work your way through it!

solution:derive f
$$9\,y \left( yg \left( 7\,{x}^{2}-9\,{y}^{2} \right) +14\,{x}^{2}y \mbox {D} \left( g \right) \left( 7\,{x}^{2}-9\,{y}^{2} \right) \right) +7\,x \left( xg \left( 7\,{x}^{2}-9\,{y}^{2} \right) -18\,x{y }^{2}\mbox {D} \left( g \right) \left( 7\,{x}^{2}-9\,{y}^{2} \right) \right)$$
gives:
$$9\,{y}^{2}g \left( 7\,{x}^{2}-9\,{y}^{2} \right) +7\,{x}^{2}g \left( 7 \,{x}^{2}-9\,{y}^{2} \right)$$
since $$f= xyg \left( 7\,{x}^{2}-9\,{y}^{2} \right)$$
we can get an expretion for y*g and x*g thus the final answer is
$$9\,{\frac {yf}{x}}+7\,{\frac {xf}{y}}$$

thanks for all the help