let x be a differentiable function x:[a,b]→R which satisfies:dx(t)dt=f(t,x(t))x(a)=xaIn particular I am trying to understand...

Sam Longoria

Sam Longoria



let x be a differentiable function x:[a,b]R which satisfies:
In particular I am trying to understand what f(t,x(t)) means, I understand this represents a multivariable function with parameters t and x(t), but I can't think of what this would mean in the given context?

Answer & Explanation




2022-01-07Added 29 answers

Think of it as if you have any first order differential equation, say:
What you are looking for is an equation x=x(t) that satisfies the given differential equation, because the analytical solution may not exist or is simply to hard or tedious to obtain. The numerical techniques give an approximation of this function in various values in an interval [ta,tb]. To calculate the approximation, we write the differential equation in the form
So, what we are doing is simply isolating in the ode the x term:
And then we have an equation that we can use numerical methods to calculate some approximations for x(t) in [ta,tb].

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