Thm: Suppose that both the function f(x,y) and its partial Derivative fy(x,y) are continuous on some rectangle R in the xy plane that contains the point (a,b) in its interior. Then, for some open interval I containing the point a, the initial value problem has one and only one solution that is defined on the interval I.
I have to determine if the above Thm does or does not guarantee existence of a solution of:
dy/dx = x lny ; y(1) = 1
Can someone show how this is done? Thanks a lot!!
The proposed DE has something 'not fully clear' because if we change the roles of x and y it becomes...
, (1)
In that case the conditions of existence of a solution are not satisfied because has a singularity in y=1. That 'anomaly' is confirmed by the fact that is we try to solve (1) we obtain as general solution...
(2)
... and it is well known that the function has a 'pole' in z=1... May be that all that requires further examination...
Kind regards
The 'thriller' about the DE...
, (1)
... has been 'solved': the solution is [simply...] . The solution cannot be derived from the 'general solution' with an appropriate value of the 'arbitrary constant' and it is called 'singular solution'...
Kind regards