ODE initial values and continuity

[mathman44]

Homework Statement

Find a continuous y(t) for t > 0 to the initial value prob:

$$y'(t)+p(t)y(t)=0, y(0)=1$$
where
$$p(t)=2$$ for 0 < t < 1
$$p(t)=1$$ for t > 1

and determine if the soln is unique.

The Attempt at a Solution

By standard ODE techniques I arrive at

$$y=\exp(-2t)$$ for 0 < t < 1
$$y=\exp(-t)$$ for t > 1

The problem is that this soln y(t) isn't continuous.. what's wrong here? As far as I know the only way to do this is to solve for y(t) in both intervals of t.

 

[mathman44]

Anyone please?

 

[Mark44]

One problem with this is that p(t) isn't defined at 0, yet your initial condition is y(0). So your differential equation isn't defined at 0, but you are supposed to find a solution y(t) that is defined at 0.

 

[mathman44]

Oops... I should have said that p(t) is 2 for t [0,1].

 

[Mark44]

Then y = e^-2t is a solution that is continuous on [0, 1], the interval that contains the initial value t = 0.

I think that's what we're looking for, but your text should have a theorem about existence and uniqueness of solutions of DEs. See what that theorem has to say about this situtation.

 

[mathman44]

But the question is asking for a continuous soln for t > 0, not just t belonging to [0,1].

 

[Mark44]

Take a look at the theorem I mentioned.