Hint:  I have not solved the whole problem but I will give the approach:

$f_{yy} - yf_{x} = 0$

Let $f = X_{x}.Y_{y}$

Then $X.Y^{''} - yY.X^{'} = 0$

$X.Y^{''} = yY.X^{'}$

$\frac{Y^{''}}{yY} = \frac{X^{'}}{X} = -\lambda$

${Y^{''}} + y\lambda Y = 0$

${X^{'}} + \lambda X = 0$

These are two ODEs in Y and X

Boundary Values are Y(0) = 0 and $\lim$ ( y tending to infinity) $Y(y) = 0$

X(x) = 0 is trivial so just leave it.

Boundary Values are used to find the general solution and the initial value is used to find the particular solution.

Initial Value f(x,0) = sin(x)

The method that I have used can be more generally summarized as follows

The Method of Separation of Variables: 


Separate the PDE into ODEs of one independent variable each. 
Rewrite the boundary conditions so they associate with only one of 
the variables. 
One of the ODEs is a part of a two-point boundary value problem. 
Solve this problem for its eigenvalues and eigenfunctions. 
Solve the other ODE. 
Multiply the results from steps (2) and (3)


Can you take it from here!!

Thanks

Satish

Analytical solution to PDE
I ran into a problem with the Eclipse formatter.  It won't format my code correctly when declaring methods within a class declaration.  It puts a new line after the method's return type.

I already exported the style xml file and examined the settings in it, but none of the settings have any apparent connection to this problem, and the settings editor in Eclipse didn't show the same problem happening in it's sample code for method declarations.

Here is an example bit of code for what I want to have happen:

class MyClass
{
    public:
        MyClass();
        void myMethod();
};


However, this is what I get:

class MyClass
{
    public:
        MyClass();
        void
        myMethod();
};


Again, in the styles editor, the code doesn't have this problem and looks just how I want it to, but in the actual code, the story is different.

I'm using version 3.8.0.  Any help is appreciated.

Edit: I deleted those source files that were formatted incorrectly (after formatting the code several times to no avail) and replaced them with "identical" files with the same methods, same structure, etc.  I formatted the code this time and it worked.  This is probably a bug, but I'm leaving it up just in case anyone else encounters a similar problem or has a solution to avoiding this problem in the first place.
