Non-Uniqueness of Solution to ODE with y(0)=0

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In summary, the conversation discusses finding the solution of a specific ODE and proving its non-uniqueness when given specific initial conditions. The conversation also includes mentioning the use of Picard's theorem and requesting a technique or explanation on how to solve nonlinear ODEs. The expert summarizer provides a summary of the solution process and expresses gratitude for the help.
  • #1
Logik
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Homework Statement



I have to find the solution of (1) and show that it is not unique if y(0) = 0.
I can prove it is not unique by using Picard's theorem but I don't know how to find the non trivial solution.

Homework Equations



(1) y(t)' = Sqrt(y(t))

The Attempt at a Solution



I don't know where to start... We have not seen how to solve nonlinear ODE's. A link to a technique or explanation to how to solve it would be very helpful. I'm not looking for the answer, I can get it with Mathematica... I want to understand how to get there.
 
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  • #2
You can directly integrate that function:

dy/dt = y^1/2 => y^(-1/2) dy = dt

Nontrivial solution. However, you'll find the trivial y(t) = 0 is a perfectly good solution to those initial conditions as well.
 
  • #3
wow I'm so stupid...

dy/dt = y^(1/2)
dy/y^(1/2) = dt
2y^(1/2) = t + c
y^(1/2) = 2t + 2c
y = 4t^2 + 8tc + c^2

thanks
 
Last edited:

1. What is the Non-Uniqueness of Solution to ODE with y(0)=0?

The Non-Uniqueness of Solution to ODE (ordinary differential equation) with y(0)=0 refers to the fact that there can be more than one possible solution to the equation that satisfies the given initial condition of y(0)=0.

2. Why does the Non-Uniqueness of Solution to ODE with y(0)=0 occur?

This occurs because ODEs are typically solved using integration techniques, which involve finding the indefinite integral of the equation. This process introduces a constant of integration, which can result in multiple solutions for the same equation and initial condition.

3. Is the Non-Uniqueness of Solution to ODE with y(0)=0 a problem?

Not necessarily. In many cases, the existence of multiple solutions does not pose a problem and can actually provide more insight into the behavior of the system described by the ODE.

4. How can we determine which solution to use in the Non-Uniqueness of Solution to ODE with y(0)=0?

The appropriate solution to use will depend on the specific context and application of the ODE. It is important to carefully analyze and consider the behavior of each solution and choose the one that best fits the problem at hand.

5. Are there any ways to avoid the Non-Uniqueness of Solution to ODE with y(0)=0?

In some cases, it may be possible to manipulate the ODE or the given initial condition to eliminate the non-uniqueness. Additionally, using alternative numerical methods or techniques such as separation of variables can also help avoid the non-uniqueness issue.

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