How to determine if x(t) is a solution to a system x'(t)=f(x)

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In summary, a solution to a system of differential equations is a function or set of functions that satisfies all of the equations in the system. To determine if a given function is a solution, one can plug the function into the system and compare it to the function f(x) by taking its derivative. A constant function can also be a solution to a system of differential equations. Initial conditions are important in determining if a function is a solution, as they provide a starting point for the function. Other methods for determining if a function is a solution include numerical methods and graphing, but the most common and accurate method is to plug the function into the equations and check for satisfaction.
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NicolaiTheDane
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For example having looked a solution sheet, I know that ##x(t)=\sin(t)## is not a solution for any system of the form ##\dot{x}(t)=f(x)##. I assume this is rather simple, but I simply cannot get my head around why it wouldn't be. I'm guessing it has to do with the dependence on ##x## rather then ##t## but I haven't gotten anywhere.
 
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If ##x(t) = \sin(t)##, then ##x'(t) = \cos(t) = \pm \sqrt{1- \sin^2(t)} = \pm \sqrt{1 - x(t)^2}##
 
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1. How do I determine if x(t) is a solution to x'(t)=f(x)?

To determine if x(t) is a solution to x'(t)=f(x), you can plug the function x(t) into the differential equation and see if it satisfies the equation. This means that when you take the derivative of x(t) and substitute it into the equation, it should equal f(x).

2. What is the significance of x(t) being a solution to x'(t)=f(x)?

If x(t) is a solution to x'(t)=f(x), it means that it is a function that satisfies the given differential equation. This means that when you plug in x(t) into the equation, it will equal the derivative of x(t) at any given time t.

3. Can there be more than one solution to x'(t)=f(x)?

Yes, there can be more than one solution to x'(t)=f(x). This is because a differential equation can have an infinite number of solutions that satisfy the equation. Different initial conditions or values for x(t) can result in different solutions.

4. How do I check if a solution to x'(t)=f(x) is unique?

To check if a solution to x'(t)=f(x) is unique, you can use the Picard-Lindelof theorem. This theorem states that if the function f(x) is continuous and satisfies a Lipschitz condition, then the solution is unique. You can also use a phase line analysis to determine if the solution is unique.

5. Is it possible for x(t) to be a solution to x'(t)=f(x) even if it does not satisfy the initial condition?

Yes, it is possible for x(t) to be a solution to x'(t)=f(x) even if it does not satisfy the initial condition. This is because the initial condition only determines the specific solution that passes through a given point. There can be other solutions that satisfy the equation but do not pass through the given point.

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