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

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SUMMARY

The discussion centers on the non-uniqueness of solutions to the ordinary differential equation (ODE) defined by y'(t) = √y(t) with the initial condition y(0) = 0. The participant demonstrates that both the trivial solution y(t) = 0 and a non-trivial solution y(t) = 4t² + 8tc + c² exist, confirming the lack of uniqueness as per Picard's theorem. The conversation highlights the integration technique used to derive the non-trivial solution and emphasizes the importance of understanding nonlinear ODEs.

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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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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.
 
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:

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