Suppose that we use Euler\'s method to approximate the solution to the different

Question

Suppose that we use Euler's method to approximate the solution to the differential equation dy / dx = x1 / y; y(0.5) = 2. Let f(x, y) = x1 / y. We let x0 = 0.5 and y0 = 2 and pick a step size h = 0.2. Euler's method is the the following algorithm. From xn and yn, our approximations to the solution of the differential equation at the nth stage: we find the next stage by computing Xn + 1 = xn + h, yn + 1 = yn + h middot f(xn, yn). Complete the following table: n xn yn The exact solution can also be found using separation of variables. It is y(x) = Thus the actual value of the function at the point x = 1.5 y( 1.5) =

Explanation / Answer

n Xn Yn 0 0.5000 2.0000 1 0.7000 2.0500 2 0.9000 2.1183 3 1.1000 2.2033 4 1.3000 2.3031 5 1.5000 2.4160 y*dy=x*dx y^2/2=x^2/2 +k,where k is a constant y(0.5)=2 4/2=0.25/2 +k k=1.875 so y^2=x^2+3.75 y=sqrt(x^2+3.75) hence y(1.5)=sqrt(1.5^2+3.75)=2.4495

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