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- Homework Statement
- 1. For which initial points exists a solution in some interval containing x-naught?

2. For which initial points exists a unique solution in some interval containing x-naught?

- 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

I know that existence means that if f is continuous on an open rectangle that contains (x

Here is my attempt at a solution for the questions:

1. y' = 6x

2. Since f

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}(y-1)^{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}(y-1)^{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}/(y-1)^{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.