1. ## [SOLVED] Please solve this...

Form the differential equation :-

'v = f( x - by) + g( x + by), where f and g are twice differentiable'.

What hint does the statement 'where f and g are twice differentiable' give us?

And the answer is in the form of Partial Derivatives. How do I know where to use Ordinary derivative and where to use the Partial Derivative?

2. Originally Posted by Altair
Form the differential equation :-

'v = f( x - by) + g( x + by), where f and g are twice differentiable'.

What hint does the statement 'where f and g are twice differentiable' give us?

And the answer is in the form of Partial Derivatives. How do I know where to use Ordinary derivative and where to use the Partial Derivative?
$\displaystyle v = f( x - by) + g( x + by)$

so:

$\displaystyle v_x = f'( x - by) + g'( x + by)$

$\displaystyle v_y = -bf'( x - by) + bg'( x + by)$

Then:

$\displaystyle v_{xx} = f''( x - by) + g''( x + by)$

and

$\displaystyle v_{yy} = b^2f''( x - by) + b^2g''( x + by)$

Hence:

$\displaystyle v_{yy}=b^2v_{xx}$

which is the 1-D wave equation

RonL

3. In the last equation after hence, shouldn't it be V instead of x? and my question still remains un answered that how will I know that partial differentiation was involved here?

4. Originally Posted by Altair
In the last equation after hence, shouldn't it be V instead of x? and my question still remains un answered that how will I know that partial differentiation was involved here?
Yes.

RonL

5. Originally Posted by Altair
my question still remains un answered
It still does.

6. Originally Posted by Altair
In the last equation after hence, shouldn't it be V instead of x? and my question still remains un answered that how will I know that partial differentiation was involved here?
1. because the question is in a section on on partial derivatives.

2. because you have functions of more than one variable

RonL

7. Thanks a lot.