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

y' = x/y, y(0.5) = 2 n = 0, x = 0.5, y(0.5) = 2 n = 1, x = 0.7, y(0.7) = y(0.5)+(0.2*0.7*(1/y(0.5))) = 2.070 n = 2, x = 0.9, y(0.9) = y(0.7)+(0.2*0.9*(1/y(0.7))) = 2.157 n = 3, x = 1.1, y(1.1) = y(0.9)+(0.2*1.1*(1/y(0.9))) = 2.259 n = 4, x = 1.3, y(1.3) = y(1.1)+(0.2*1.3*(1/y(1.1))) = 2.374 n = 5, x = 1.5, y(1.5) = y(1.3)+(0.2*1.5*(1/y(1.3))) = 2.500 dy/dx = x/y =>y^2/2 = x^2/2 + c y(0.5) = 2 => 2 = 0.25/2+c => c = 1.875 =>y^2 = x^2+3.75 y(1.5) = 2.449489743

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