I PDE with recursive relation

1. Mar 28, 2017

MAGNIBORO

hi, I do not know much about PDEs and programs like wolfram alpha and maple dont give me a solution.
it is possible to calculate the function through PDE?.
I would appreciate any help

$$\frac{\partial }{\partial a}f(a,b,n)+\frac{\partial }{\partial b}f(a,b,n)=-n f(a,b,n+1)$$

$$f(a,b,0)=\frac{\pi }{2} \: \; \, \,, \, \, \, f(a,b,1)=\frac{\pi }{2\sqrt{a b}}$$

as we know $f(a,b,1)$ We can calculate $f(a,b,2)\,,\,f(a,b,3),...$

But we could calculate closed form expression for $f(a,b,n)$ ?

thanks

2. Mar 28, 2017

MAGNIBORO

i reduce the problem to solve
$$\frac{\partial }{\partial c} \, g_{n}(c) = -n \, g_{n+1}(c)$$
with
$$g_{1}(c)=\frac{\pi }{2\sqrt{c^{2}+c}}$$

3. Apr 14, 2017

Strum

Do you need the result for all c?

4. Apr 15, 2017

MAGNIBORO

i found the solution, The function was the result of a definite integral.
thanks anyway

5. Apr 17, 2017

Strum

Well if you found the solution I would like to see it. Please post it :)

6. Apr 17, 2017

MAGNIBORO

the function is:
$$\int_{0}^{\frac{\pi }{2}}\left ( a \, cos^2(x) +b \, sin^2(x) \right )^{-n}$$

7. Apr 19, 2017

Strum

Ehh but this is not dependent on c as specified in post number 2.

8. Apr 19, 2017

MAGNIBORO

evaluate the function as $f(c,c+1,n)$ and check