Analytical solution to PDE I am trying to solve the following linear pde where $u=f(x,y)$ in the domain $y \in  (0,\infty)$: 
$$y\dfrac{\partial{u}}{\partial x} = \dfrac{\partial^2 u}{\partial y^2}$$
with boundary conditions: 
$$u(x,0)=\sin(x) $$ 
$$\lim_{y \rightarrow \infty} u(x,y) = 0 .$$
Can someone please suggest how do I proceed to get the analytical solution for this equation? 

Solution: 
Using method of separation of variables, 
we assume solution to be of the form: $$u(x,y)=X(x).Y(y) $$
Inserting this to the pde, we get: 
$$yX'Y=XY'' $$
$$\dfrac{X'}{X}=\dfrac{Y''}{yY}=-\lambda$$
Now we get the following ode's 
$$X'+\lambda X=0 $$
$$Y'' + \lambda y Y = 0 $$
from which we get two general solutions of the following form: 
$$X(x) = c_1\exp(-\lambda x) \qquad Y(y) = c_2Ai(\lambda^{1/3} y) + c_3Bi(\lambda^{1/3} y)  $$ where $Ai$ and $Bi $ are Airy's functions and the general solution can be expressed as: 
$$u(x,y) =  c_1\exp(-\lambda x).(c_2Ai(\lambda^{1/3} y) + c_3Bi(\lambda^{1/3} y))$$
$c_3 \rightarrow 0 $ to satisfy second boundary condition since $Bi(\infty) \rightarrow \infty $ and hence, solution is of the form: 
$$u(x,y) = A \exp(-\lambda x) Ai(\lambda^{1/3} y) $$
Putting $y=0$, 
$$u(x,0) = A\exp(-\lambda x).\dfrac{1}{3^{2/3}}\Gamma({2/3}) = \sin(x) $$
How should I now proceed to find $A$ and $\lambda$, considering that $\lambda$ needs to be positive real value?
 A: 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
A: EDITED (to change cos to sin)
Solution seems to be
$$\eqalign{\dfrac{\Gamma(2/3)\; 3^{2/3}}{8} & \left((-\sqrt{3}+i) e^{-ix} Ai(-(\sqrt{3}+i)y/2) \right.\cr
&  - (\sqrt{3} + i) e^{ix} Ai(-(\sqrt{3}-i)y/2) \cr
&  + (\sqrt{3} i + 1) e^{-ix} Bi(-(\sqrt{3}+i)y/2)\cr
& + \left. (-\sqrt{3} i + 1) e^{ix} Bi(-(\sqrt{3}-i)y/2)   \right)\cr}
$$
where $Ai$ and $Bi$ are Airy functions.
