Help - Seperation of variables problem, multiple solutions.

In summary, the given problem involves finding the solution to the differential equation dy/dx = √y with initial condition y(0) = 0. There are infinitely many solutions, and three of them are y=0, y=x^2/4, and y= (x-2)^2/4 for x≥2. The key to finding these solutions is to consider the behavior of the derivative when y=0.
  • #1
girlphysics
43
0
Help -- Seperation 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



Seperation 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.
 
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  • #2
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.
 
  • #3
girlphysics said:

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



Seperation 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! :smile:

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

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



Seperation 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 [itex]y' = \sqrt y = 0[/itex] for all [itex]x[/itex].
This suggests something like
[tex]
y(x) = \left\{\begin{array}{r@{\quad}l}
\frac{(x - a)^2}{4}, & x < a \\
0, & a \leq x \leq b \\
\frac{(x - b)^2}{4}, & x > b\end{array}\right.[/tex]
for [itex]a < 0 < b[/itex].
 
  • #5
I like Serena said:
Hey you! :smile:

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?
 
  • #6
girlphysics said:
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. :blushing:
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##.
 
Last edited:
  • #7
If you're still interested, you can find more information on your problem http://www.mathhelpboards.com/f17/interesting-ordinary-differential-equation-3684/.
 
  • #8
I like Serena said:
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.
 

1. What is the "separation of variables" method?

The separation of variables method is a mathematical technique used to solve partial differential equations. It involves assuming that the solution can be written as a product of two functions, each depending on only one of the independent variables.

2. How does the separation of variables method work?

The separation of variables method involves breaking down a partial differential equation into simpler ordinary differential equations by separating the dependent variables and solving each equation individually. The solutions are then combined to form the overall solution.

3. Why are there multiple solutions when using the separation of variables method?

The separation of variables method can sometimes result in multiple solutions because it involves breaking down a complex equation into simpler parts. These individual solutions may not necessarily be unique, leading to multiple possible solutions for the original equation.

4. How do you determine which solution is the correct one?

Determining the correct solution when using the separation of variables method is dependent on the initial conditions or boundary conditions of the problem. These conditions can be used to select the appropriate solution that satisfies the given constraints.

5. What are some common applications of the separation of variables method?

The separation of variables method is commonly used in physics, engineering, and other sciences to solve partial differential equations that arise in various fields, such as heat transfer, fluid dynamics, and quantum mechanics.

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