Question

For each initial value problem, determine whether Picard's Theorem can be used to show the existence of a unique solution in an open interval containing t = 0. Justify your answer.

(a) y' = ty4/3, y(0) = 0
(b) y' = tył/3, y(0) = 0
(c) y' = tył/3, y(0) = 1

Answer 1

Part a:
`f , \, f_y` is continuous at the initial value (0,0) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.

Part b:
`f_y` is not continuous at the initial value (0,0) so due to Picardi theorem there does not exist an interval such that the IVP has a unique solution.

part c:
`f , \, f_y` is continuous at the initial value (0,1) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.

Explanation:

Part a

as
`y^(' )=ty^(4/3)`

Let

`f(t,y)=ty^(4/3)`

Now derivative wrt y is given as

`f_y=(4)/(3)ty^(1/3)`

Finding continuity via the initial value

`f` is continuous on
`R^2` also
`f_y` is also continuous on
`R^2`

Also

`f , \, f_y` is continuous at the initial value (0,0) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.

Part b

as
`y^(' )=ty^(1/3)`

Let

`f(t,y)=ty^(1/3)`

Now derivative wrt y is given as

`f_y=(1)/(3)ty^(-2/3)`

Finding continuity via the initial value

`f` is continuous on
`R^2` also
`f_y` is also continuous on
`R^2`

Also

`f_y` is not continuous at the initial value (0,0) so due to Picardi theorem there does not exist an interval such that the IVP has a unique solution.

Part c

as
`y^(' )=ty^(1/3)`

Let

`f(t,y)=ty^(1/3)`

Now derivative wrt y is given as

`f_y=(1)/(3)ty^(-2/3)`

Finding continuity via the initial value

`f` is continuous on
`R^2` also
`f_y` is also continuous on
`R^2` when
`y\\eq 0`

Also

`f , \, f_y` is continuous at the initial value (0,1) so due to Picardi theorem there exists an interval such that the IVP has a unique solution.