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Solution of Differential Equations

  1. Runge-Kutta method
     
    The rule for finding the increment k of y corresponding to the increment h of x.
     
    Description: 448.png= f (x, y) ; y (x0) = y0
     
    Calculation is done as follows :
     
    k1 = h f (x0, y0), k2 = Description: 453.png
     
    k3 = Description: 458.png, k4 = hf (x0 + h, y0 + k3)
     
    k = Description: 463.png (k1 + 2k2 + 2k3 + k4) =Description: 468.png (k1 + k4) + Description: 473.png (k2 + k3)
     
    The approximate value, y1 = y+ k
  2. Taylor’s series method
     
    Given Description: 478.png = y = f (x, y) and f (x0) = y0
     
    By taylor’s theorem the series about a point x = x0
     
    y = y0 + (x  x0) (y)0 + Description: 483.png
     
    From this equation we can find y1 of y for x = x1 and then y, y, y″′ are found out.
  3. Euler’s method
     
    Given Description: 488.png = y = f (x, y) and y(x0) = y0
     
    when h  0, by Taylor’s series
     
    y(x + b) = y (x) + h y(x) = y(x) + h f (x, y)
     
    Thus, yn+1 = yn h f( xn, yn)
     
    where, h = Description: 493.pngi.e. (xn = x0 + nh)




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