# let x be a differentiable function x:[a,b] \rightarrow R which

let x be a differentiable function $$\displaystyle{x}:{\left[{a},{b}\right]}\rightarrow{R}$$ which satisfies:
$$\displaystyle{\frac{{{\left.{d}{x}\right.}{\left({t}\right)}}}{{{\left.{d}{t}\right.}}}}={f{{\left({t},{x}{\left({t}\right)}\right)}}}$$
$$\displaystyle{x}{\left({a}\right)}={x}_{{a}}$$
In particular I am trying to understand what $$\displaystyle{f{{\left({t},{x}{\left({t}\right)}\right)}}}$$ means, I understand this represents a multivariable function with parameters $$\displaystyle{t}$$ and $$\displaystyle{x}{\left({t}\right)}$$, but I can't think of what this would mean in the given context?

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turtletalk75
Think of it as if you have any first order differential equation, say:
$$\displaystyle{\left({3}{x}^{{2}}+{1}\right)}{t}{x}=-{x}^{{3}}+{x}$$
What you are looking for is an equation $$\displaystyle{x}={x}{\left({t}\right)}$$ 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 $$\displaystyle{\left[{t}_{{a}},{t}_{{b}}\right]}$$. To calculate the approximation, we write the differential equation in the form
$$\displaystyle{x}{\left({t}\right)}={f{{\left({t},{x}{\left({t}\right)}\right)}}}$$
So, what we are doing is simply isolating in the ode the x term:
$$\displaystyle{x}={\frac{{-{x}^{{3}}+{x}}}{{{\left({3}{x}^{{2}}+{1}\right)}{t}}}}$$
And then we have an equation that we can use numerical methods to calculate some approximations for $$\displaystyle{x}{\left({t}\right)}$$ in $$\displaystyle{\left[{t}_{{a}},{t}_{{b}}\right]}$$.