ghost_writer
New member
- Joined
- Feb 10, 2010
- Messages
- 1
Question: Find the interval of existence of the following:
y ' = 2(1 - t^2 - y^2)^(1/2); y(0) = 0
I don't understand how my prof explains how to apply the existence and uniqueness theorem for DEs here. I know you have to find where f(t,y) and df/dy (partial derivative of f w/ respect to y) are both continuous first:
f(t,y) is continuous for t^2 + y^2 <= 1
df/dy =(1 - t^2 - y^2)^(-1/2)(2y) so continuous for t^2 + y^2 < 1
But I'm not sure what my prof is doing after that. It has something to do with how the function is symmetric or f(t,y) = f(y,t) and the definition h >= min{a, a/m} where m = max|f| and you need to solve for a to maximize h using right triangles, but my prof frequently switches a's and b's in this definition for the region (-a, a) x (-b, b). I'm just completely lost after checking the continuity.
I've been searching online for problems that match this but most resources go right to series approximation for this theorem w/o using the application above. Any help to understanding this is much appreciated.
y ' = 2(1 - t^2 - y^2)^(1/2); y(0) = 0
I don't understand how my prof explains how to apply the existence and uniqueness theorem for DEs here. I know you have to find where f(t,y) and df/dy (partial derivative of f w/ respect to y) are both continuous first:
f(t,y) is continuous for t^2 + y^2 <= 1
df/dy =(1 - t^2 - y^2)^(-1/2)(2y) so continuous for t^2 + y^2 < 1
But I'm not sure what my prof is doing after that. It has something to do with how the function is symmetric or f(t,y) = f(y,t) and the definition h >= min{a, a/m} where m = max|f| and you need to solve for a to maximize h using right triangles, but my prof frequently switches a's and b's in this definition for the region (-a, a) x (-b, b). I'm just completely lost after checking the continuity.
I've been searching online for problems that match this but most resources go right to series approximation for this theorem w/o using the application above. Any help to understanding this is much appreciated.