Simple ODE uniqueness/domain question

[bmxicle]

Homework Statement

Find the solution satisfying the given initial conditions for dy/dx = y/x, y(0) = 0 and explain according to the existence and uniqueness theorem.

Homework Equations

Existence/uniqueness theorem for non-linear ODE

The Attempt at a Solution

I'm mainly just having a hard time figuring out the interval of validity, and the interpretation of the existence/uniqueness theorem.

For the general solution: dy/dx = y/x ====> y(x) = cx
y(0) = 0 = c(0) ==> c = 0

Giving Y(x) = 0 for this initial value.

The existence/uniqueness theorem does not guarantee that a unique solution exists for y(0) = 0 since f(x,y) = x/y is not continuous for y = 0.

Is this solution valid even though dy/dx is undefined for the intial value. I'm also confused as to whether this solution should be valid over all real values of x or not.

 

[mighty2000]

Thank you for your post. The solution you have found is valid for all real values of x. This is because the existence and uniqueness theorem for non-linear ODEs states that if the function f(x,y) is continuous in a region R of the xy-plane, then there exists a unique solution that satisfies the initial condition at any point (a,b) in R. In this case, the function f(x,y) = y/x is continuous for all real values of x except for x = 0. Therefore, the solution is valid for all real values of x except for x = 0.

However, since the initial condition given is y(0) = 0, the solution is not valid at x = 0. This is because the function f(x,y) is not continuous at this point, and thus the existence and uniqueness theorem does not apply.

In summary, the solution you have found is valid for all real values of x except for x = 0, but it is not valid at x = 0 due to the discontinuity of the function at this point. I hope this helps to clarify the interpretation of the existence and uniqueness theorem in this case. Keep up the good work with your studies!