Estimate the differential equation: y' = y – x – 1, y(0) = 1 Estimate y(1) using a step size h = 0.5 a) Use Euler's Method to estimate y(1) Euler's Method Yn+1 = Yn + hf(xn, Yn) |f(Xn, Ka) Yn+1 1 0.5 0.5 0.5 Improved Euler's Method Yn+1 = Yn +F (xn, Yn) + f(xn+1+Yn+1) Yn+1 = Yn + hf (xn, Yn) b) Use Improved Euler's Method to estimate y(1) h f(Xa. Ka) |f(Xa. Ka") Yn+1 1 0.5 0.5 0.5

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.