What the test question that just bugged the crap outta me

[schattenjaeger]

we were given the DE y'=2sqrt(y) that was how it was given, I didn't take the square root myself and thus forget a +and-

so using a bernoulli equation I got (x+c)^2(I think, I'm doing this from memory)

then the next part asked to find two solution curves for the point (1,1), so x^2 and (x-2)^2, right?

THEN it asked for the solution guaranteed by the existence and uniqueness theorem at (1,1), but didn't I just find TWO solution curves for that same point, which means there ISN'T an unique solution? And to further confuzzle matters if I actually apply the theorem it seems like it SHOULD have an unique solution at (1,1)

 

[quantumdude]

Take a closer look at the solution $y=(x-2)^2$. It has a negative slope at $x=1$. If $y'<0$, then according to your DE $2\sqrt{y}<0$, which is impossible.

 

[schattenjaeger]

aw hell, so when it asked me to sketch two solution curves THAT was the "trick" question?

 

[schattenjaeger]

or were they both solution curves but they passed through the point or something? We'll go over it on Monday but I hate waiting to know

 

[quantumdude]

They are not both solutions at $x=1$. $y=(x-2)^2$ simply does not satisfy the differential equation at $x=1$. What happened was that you introduced an extraneous root when you squared $y^{1/2}$.