 #1
ver_mathstats
 260
 21
 Homework Statement:

1. For which initial points exists a solution in some interval containing xnaught?
2. For which initial points exists a unique solution in some interval containing xnaught?
 Relevant Equations:
 Existence and Uniqueness
I'm new to learning about ODE's and I just want to make sure I am on the right track and understanding everything properly.
We have our ODE which is y' = 6x^{3}(y1)^{1/6} with y(x_{0})=y_{0}.
I know that existence means that if f is continuous on an open rectangle that contains (x_{0}, y_{0}) then the IVP has at least one solution on some open subinterval of (a,b) that contains x_{0}. Uniqueness is when f and f_{y} are continuous on the rectangle then we will have a unique solution on some open subinterval of (a,b) that contains x_{0}.
Here is my attempt at a solution for the questions:
1. y' = 6x^{3}(y1)^{1/6} is continuous for all x∈R and all y∈R, thus there is a solution for all (x_{0}, y_{0}) according to the theorem of existence.
2. Since f_{y} = x^{3}/(y1)^{5/6} is continuous for all x∈R and all y∈R except for y = 1 we see there is a unique solution on some open interval containing x_{0} for all (x_{0}, y_{0}) except when y = 1.
Could someone please check this over to see if I have the right idea and if this is correct? Thank you.
We have our ODE which is y' = 6x^{3}(y1)^{1/6} with y(x_{0})=y_{0}.
I know that existence means that if f is continuous on an open rectangle that contains (x_{0}, y_{0}) then the IVP has at least one solution on some open subinterval of (a,b) that contains x_{0}. Uniqueness is when f and f_{y} are continuous on the rectangle then we will have a unique solution on some open subinterval of (a,b) that contains x_{0}.
Here is my attempt at a solution for the questions:
1. y' = 6x^{3}(y1)^{1/6} is continuous for all x∈R and all y∈R, thus there is a solution for all (x_{0}, y_{0}) according to the theorem of existence.
2. Since f_{y} = x^{3}/(y1)^{5/6} is continuous for all x∈R and all y∈R except for y = 1 we see there is a unique solution on some open interval containing x_{0} for all (x_{0}, y_{0}) except when y = 1.
Could someone please check this over to see if I have the right idea and if this is correct? Thank you.