18.03 Lecture Notes - Lecture 2: Euler Method, Slope Field, Numerical Analysis
Document Summary
[1] the study of differential equations has three parts: Even if we can solve symbolically, the question of computing values remains. The number e is the value y(1) of the solution to y" = y with y(0) = 1. The answer is: numerical methods. (euler already computed e to at least 18 decimal places. It is now known to some 200 billion places, but i won"t write them all out here. ) As an example, take the first order ode y" = y^2 - x = f(x,y) with initial condition y(0) = -1. I invoked the euler"s method mathlet, and selected f(x,y) = y^2 - x . I selected the initial condition (0,-1) , and then invoked "actual. " This solution is one of those trapped in the funnel, so for large x , the graph of y(x) is close to the graph of. \sqrt(x) : y(100) is very close to -10 . [2] the tangent line approximation gives one approach.
