(a) A 3rd order Runge-Kutta method for solving the differential equation y' = f(x, y) with...

 

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(a) A 3rd order Runge-Kutta method for solving the differential equation y' = f(x, y) with initial condition x = xo, y = yo is given by, k = hf (xn, yn); k2=hf (xn+h/2, yn+k1/2); k3=hf (xn+h, yn + 2k2 - k); Yn+1 yn+(k+4k2 + k3). = Apply this method to calculate approximate solutions to the differential equation, yry x, with x = 0, y = 2 at the point where x = 0.2, using h= 0.1. Work to 6 decimal places throughout and state your final solution to 5 decimal places. (b) Applying the 3rd order Runge-Kutta method using h = 0.2, the global error for a differ- ential equation in the y solution at x = 0.4 is found to equal 0.310401. If the solution at the same point was recalculated using h= 0.05 approximately what would you expect to happen to the global error and how many decimal places of accuracy would you expect in the new y solution? (c) Use the Taylor series, h h y(x + h) = y(x) + hy'(x) + y ) + hy '(x) + 12 + y (x) + 1/2+2) + to find the solution to the same differential equation and initial condition from Part (a) at the point x = 0.2 using = 0.2. Use 7 terms in the series, work throughout to 6 decimal places and state the final solution to 5 decimal places. (a) A 3rd order Runge-Kutta method for solving the differential equation y' = f(x, y) with initial condition x = xo, y = yo is given by, k = hf (xn, yn); k2=hf (xn+h/2, yn+k1/2); k3=hf (xn+h, yn + 2k2 - k); Yn+1 yn+(k+4k2 + k3). = Apply this method to calculate approximate solutions to the differential equation, yry x, with x = 0, y = 2 at the point where x = 0.2, using h= 0.1. Work to 6 decimal places throughout and state your final solution to 5 decimal places. (b) Applying the 3rd order Runge-Kutta method using h = 0.2, the global error for a differ- ential equation in the y solution at x = 0.4 is found to equal 0.310401. If the solution at the same point was recalculated using h= 0.05 approximately what would you expect to happen to the global error and how many decimal places of accuracy would you expect in the new y solution? (c) Use the Taylor series, h h y(x + h) = y(x) + hy'(x) + y ) + hy '(x) + 12 + y (x) + 1/2+2) + to find the solution to the same differential equation and initial condition from Part (a) at the point x = 0.2 using = 0.2. Use 7 terms in the series, work throughout to 6 decimal places and state the final solution to 5 decimal places.