Given y'=-y with y(0)=1 determine the value of y at x=0.01,0.02 by Eule's method.

Given y'=-y with y(0)=1 determine the value of y at x=0.01,0.02 by Eule's method.

Solution

             

        Given,    dy/dx=-y.        ,.  Y(0)=1

      Compare,   dy/dx=f(x,y). ,.  Y(xo)=yo

   With.    

f(x,y)=-y. ,.    Xo=0    ,        yo=1

                   X1=0.01 ,.        y1=?

                   X2=0.02 ,.        y2=?

To find y(0.01)=y1

           

h = x1 - xo       

h  = 0.01 - 0     

 h = 0.01           

f(x,y) =  - y

f(0,1) = - 1

    y1=y(0.01)= yo + h  f(xo,yo)

                       = 1 + 0.01 f(0,1)

                       = 1+ 0.01 (-1)

                        =  1 - 0.01

                   y1 = 0.99

To find y(0.02) = y2

h = x2 - x1

h = 0.02 - 0.01

h = 0.01

f(x, y).     = - y

f(X1,y1) =  f(0.01,0.99)

f(X1,y1) = - 0.99

   y2 = y (0.02) = y1 + h f( X1,X2)

                         = 0.99 +0.01  (0.01,0.99)

                          = 0.99 + 0.01(-0.99)

                    y2  = 0.9801