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## Homework Statement

So yeah, my first time playing with ODEs, how exciting. So my prof gave us a few suggested exercises and I want to know whether I'm actually doing these properly or not. The question and all relevant things will be included in the picture below :

http://gyazo.com/b3be5b5d56ce2201cf88bce5ea8d5838

## Homework Equations

## The Attempt at a Solution

(1a) So! I figured I should probably break this up into cases since I'm dealing with a piecewise function.

Case #1 : Suppose x>c, then y(x) = (x-c)^2 and y' = 2(x-c). Now we check that this satisfies our equation!

y' = 2[itex]\sqrt{y}[/itex]

2(x-c) = 2[itex]\sqrt{(x-c)^2}[/itex]

2(x-c) = 2(x-c)

1 = 1

Case #2 : Suppose x ≤ c, then y(x) = 0 and y' = 0. Now we check that this also satisfies our equation.

y' = 2[itex]\sqrt{y}[/itex]

y' = 2[itex]\sqrt{0}[/itex]

0 = 0

Check and check. ∴ y(x) satisfies our ODE for all x as desired.

Now for the figure, I've never drawn a direction field before, so I got the computer to do it for me for the first time. I know of course that the equilibrium point occurs at y' = 0 ( In this case it turns out after solving we get y = 0 ). Here's a picture of the direction field when we have the IVP y(0) = 0 :

http://gyazo.com/443b52dec28b53e61b3c7ffb3c0b4369

I'm not sure how this demonstrates that we have infinitely many solutions, perhaps someone could elaborate for me?

I'll save my attempt at (1b) for after I know what I'm doing with part (1a).