This justifies our attempt to use numerical methods

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can prove it exists and that it satisfies the IVP. This justifies our attempt to use numerical methods to approximate y ( x ) . Defining the Picard iterates Recall that we are assuming our differential equation is of the form dy dx = f ( x, y ) . It will be helpful to rewrite this differential equation as an integral equation by integrating both sides from x 0 to x to get y ( x ) - y 0 = Z x x 0 f ( t, y ( t )) dt (1) y ( x ) = y 0 + Z x x 0 f ( t, y ( t )) dt We can think of the right hand side of equation (1) as an operator on functions. That is, it takes in a function y ( x ) as an input, and produces another function as an output. If we call this operator L , then we can write this idea down as follows: L [ y ( x )] = y 0 + Z x x 0 f ( t, y ( t )) dt.
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In words, what the operator L does to the function y ( x ) is: (i) Takes an input y ( x ) and forms the new function f ( t, y ( t )) in the dummy variable t . (ii) Integrates this new function from x 0 to x , thus producing a function of x : Z x x 0 f ( t, y ( t )) dt (iii) Adds the constant y 0 , producing the final result, which is a function of x : y 0 + Z x x 0 f ( t, y ( t )) dt Now look again at the integral equation (1). What this equation is saying is that the solutions y ( x ) to our initial value problem are precisely
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