Using the Euler's method (with h = 10 − n ) to find y ( 1 ) y ′ = sin ⁡ ( x ) x Since y ′ ( x ) = sin ⁡ ( x ) x then f ( x , y ) = sin ⁡ ( x ) x I know y n + 1 = y n + h ⋅ f ( x n , y n )and given y ( 0 ) = 0, so x 0 = y 0 = 0Therefore, x 1 = x 0 + h = 0 + 10 − 0 = 1 ⇒ y 1 = y ( x 1 ) = y ( 1 )Then, y 1 = 0 + 10 − 0 ⋅ f ( x 0 , y 0 ) y 1 = f ( 0 , 0 )But f ( 0 , 0 ) = sin ⁡ ( 0 ) 0 = 0 0 How am I supposed to do it?

Question

Using the Euler&#039;s method (with   h  =      10          −      n      ) to find   y  (  1  )      y    ′    =            sin      ⁡      (      x      )        x  Since      y    ′    (  x  )  =            sin      ⁡      (      x      )        x  then  f  (  x  ,  y  )  =            sin      ⁡      (      x      )        x  I know      y          n      +      1        =      y          n        +  h  ⋅  f  (      x          n        ,      y          n        )and given   y  (  0  )  =  0, so      x          0        =      y          0        =  0Therefore,      x          1        =      x          0        +  h  =  0  +      10          −      0        =  1  ⇒      y          1        =  y  (      x          1        )  =  y  (  1  )Then,      y          1        =  0  +      10          −      0        ⋅  f  (      x          0        ,      y          0        )      y          1        =  f  (  0  ,  0  )But  f  (  0  ,  0  )  =            sin      ⁡      (      0      )        0    =      0    0  How am I supposed to do it?

Alden Holder · Accepted Answer

You have to use boundary condition   y  (  0  )  =  1 (i.e. consistent boundary condition) to keep the right hand side continuous. In other words, you have to solve a bit different problem                              y          ′                (        x        )        =                              {                                                                                                           1                                                                                                                                      if                             x              =              0                                                                                            sin                  ⁡                  x                                x                                                                                                                      otherwise                                          Otherwise it is not solvable. (The right hand side must be defined in the starting point 0, where             sin      ⁡      x        x   is not.)

Using the Euler's method (with h=10^(−n)) to find y(1) y′=sin(x)/x

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