Help - Seperation of variables problem, multiple solutions.

[girlphysics]

Help -- separation of variables problem, multiple solutions.

Homework Statement

Suppose that dy/dx = √y and y(0) = 0. What is y(x)? There is more than one answer to this problem. You must list five correct solutions.

Homework Equations

separation of Variables/ integration

The Attempt at a Solution

I got the first solution to be y= x^2 /4 and c=0. I don't know how to get the four other solutions.

 

[CompuChip]

If you restrict a function to a subset of its domain, it is technically a different function. Could this be what is meant? Because the given IVP seems unambiguous to me.

 

[I like Serena]

Homework Statement

Suppose that dy/dx = √y and y(0) = 0. What is y(x)? There is more than one answer to this problem. You must list five correct solutions.

Homework Equations

separation of Variables/ integration

The Attempt at a Solution

I got the first solution to be y= x^2 /4 and c=0. I don't know how to get the four other solutions.

Hey you!

There must be some mistake in the problem statement.
The solution y=x^2/4 is the only one!

 

[pasmith]

Homework Statement

Suppose that dy/dx = √y and y(0) = 0. What is y(x)? There is more than one answer to this problem. You must list five correct solutions.

Homework Equations

separation of Variables/ integration

The Attempt at a Solution

I got the first solution to be y= x^2 /4 and c=0. I don't know how to get the four other solutions.

y(x) = 0 is also a solution, since then $y' = \sqrt y = 0$ for all $x$.
This suggests something like
$$y(x) = \left\{\begin{array}{r@{\quad}l}<br /> \frac{(x - a)^2}{4}, & x < a \\<br /> 0, & a \leq x \leq b \\<br /> \frac{(x - b)^2}{4}, & x > b\end{array}\right.$$
for $a < 0 < b$.

 

[girlphysics]

Hey you!

There must be some mistake in the problem statement.
The solution y=x^2/4 is the only one!

Hey! There is no mistake, my professor talked about it in class and said there are infinitely many solutions, and that he wants us to get the 3rd solution. The second solution someone posted below. He said it is difficult to get the third. any ideas?

 

[I like Serena]

Hey! There is no mistake, my professor talked about it in class and said there are infinitely many solutions, and that he wants us to get the 3rd solution. The second solution someone posted below. He said it is difficult to get the third. any ideas?

My mistake.
This turns out to be an interesting problem!
I did not realize this system had more than one solution.
I see now that they are caused because separation of variables leads to a system that is not defined for y=0, causing multiple solutions.

Anyway, I believe pasmith came up with the key to the solutions.

You can pick $y=0$ for $x<0$ and $y=x^2/4$ for $x\ge 0$.

Or you can pick $y=0$ for $x<0$, $y=0$ for $0 \le x < 2$, and $y=(x-2)^2/4$ for $x \ge 2$.
Or...

However, I believe his part of the solution for x<a is faulty, since the derivative becomes negative which does not match with $\sqrt y$.

 

[I like Serena]

If you're still interested, you can find more information on your problem http://www.mathhelpboards.com/f17/interesting-ordinary-differential-equation-3684/.

 

[girlphysics]

If you're still interested, you can find more information on your problem http://www.mathhelpboards.com/f17/interesting-ordinary-differential-equation-3684/.

Thank you so much! I finally understand. I really appreciate it.