Can I Find a Unique Solution to the Given 2nd Order Differential Equation?

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In summary, the given conversation discusses the existence and uniqueness theorem for a Cauchy problem involving a second-order linear differential equation with initial conditions. The equation does not satisfy the conditions for a unique solution at the initial point x_0 = 0 due to discontinuity in the coefficients p and q. The first question asks if this means there is no unique solution, while the second question involves finding a general solution and using initial conditions to determine the values of the constants c_1 and c_2. The expert concludes that the existence and uniqueness theorem cannot be applied in this case, but there may still be a possibility of finding a unique solution in a neighborhood of the initial point. However, it appears unlikely given the singularity at x
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Darkmisc
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I've been given a 2nd ODE in the form

y'' + p(x)y' + q(x)y = 0

The equation does not satisfy the test for a unique solution at x_0 = 0, because p and q are not continuous at x_0 (both p and q have x in the denominator, so a value of 0 makes the function discontinuous).

I've two questions.

1) If the uniqueness-existence theorem is not satisfied, can I conclude that the theorem does not guarantee a unique solution, but failure of the test does not exclude there being a unique solution? I.e. a unique solution is possible, but its existence cannot be predicted using the theorem.

2) I've found a general solution to the 2nd ODE of

y_h = c_1*f(x) + c_2*g(x)

I'm given initial conditions of y(0) =0 and y'(0) = 1.

using these values, I get c_2 = 4c_1 - 4.

The other equation I obtain cancels to 0 = 0

From this, can I conclude that no unique solution exists (for those initial values) to the 2nd ODE, since I'm not able to obtain unique values for c_1 and c_2 ?


Thanks
 
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I suppose you refer to the existence and uniqueness theorem for the Cauchy problem. If the DE does not satisfy the conditions of that theorem, then you can't apply it, but yes, this doesn't mean that you can by luck find a solution, or you can be even luckyer, and it could happen that that solution is unique. However the theorem says that you can find an unique solution in a neighbourhood of the initial point, so it is essentially a local theorem, not a global one. In your case, since the DE is singular at x = 0, it seems to me that it's quite impossible to find a solution whose domain includes that point. Unless you found a solution that contains an x factor that cancels with the x in the denominator of p and q? Ignoring this, yes, if you found two solutions then... you proved that the solution is not unique! There are many examples of this, and the DE doesn't even need to be discontinuous (the Cauchy theorem imposes also other conditions).
 

1. What is a 2nd order differential equation?

A 2nd order differential equation is a mathematical equation that involves a second derivative of a function. It is commonly used in physics and engineering to model physical systems.

2. What is the difference between a unique solution and a general solution?

A unique solution is a specific solution to a differential equation that satisfies all initial conditions, while a general solution is a family of solutions that includes all possible solutions to a differential equation.

3. How do you determine if a 2nd order differential equation has a unique solution?

A 2nd order differential equation has a unique solution if its coefficients are continuous and the equation satisfies the Lipschitz condition, which states that the slope of the solution curve must not be too steep at any point.

4. Can a 2nd order differential equation have more than one unique solution?

No, a 2nd order differential equation can only have one unique solution if it satisfies the conditions for a unique solution.

5. How are initial conditions used to determine a unique solution to a 2nd order differential equation?

Initial conditions, such as the values of the function and its first derivative at a point, are used to find the specific solution to a differential equation that satisfies those conditions. This unique solution can then be used to model the behavior of the system.

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