x ″ − t x ′ + x 2 = t x ( 0 ) = 1 x ′ ( 0 ) = 1a) Restate the problem solving a system of first-order ODEs. x 1 ′ = x 2 x 2 ′ = t − x 1 2 + t x 2 x 1 ( 0 ) = 1 x 2 ( 0 ) = 1b) Use part a) and Euler's method with h = 0.1 to find x(0.2). x ( 0.1 ) = 1 + ( 0.1 ) ( 0 − 1 + 0 ) = 0.9Is my work up to this point correct? I'm unsure whether this should actually be two equations.

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x    ″    −  t      x    ′    +      x    2    =  t  x  (  0  )  =  1      x    ′    (  0  )  =  1a) Restate the problem solving a system of first-order ODEs.      x    1    ′    =      x    2        x    2    ′    =  t  −      x    1    2    +  t      x    2        x    1    (  0  )  =  1      x    2    (  0  )  =  1b) Use part a) and Euler&#039;s method with h = 0.1 to find x(0.2).  x  (  0.1  )  =  1  +  (  0.1  )  (  0  −  1  +  0  )  =  0.9Is my work up to this point correct? I&#039;m unsure whether this should actually be two equations.

tangouwn · Accepted Answer

Your part a) is correct, but for part b) remember that you&#039;ve made       x    1    =  x. Thus finding an approximate value for   x  (  0.2  ) is equivalent to finding an approximate value for       x    1    (  0.2  ), which you would find using the first of your two equations (looks like you&#039;ve used the second one).With the first equation, we use Euler&#039;s method to obtain the approximation       x    1    (  0.2  )  ≈      x    1    (  0.1  )  +  0.1  ⋅      x    2    (  0.1  ). We don&#039;t have       x    1    (  0.1  ) and       x    2    (  0.1  ) yet, we need to approximate them using Euler&#039;s method. Using the first equation,       x    1    (  0.1  )  ≈      x    1    (  0  )  +  0.1  ⋅      x    2    (  0  )  =  1.1. Using the second equation,       x    2    (  0.1  )  ≈      x    2    (  0  )  +  0.1  ⋅  (  0  −      x    1    (  0      )    2    +  0  )  =  0.9. Thus finally,      x    1    (  0.2  )  ≈  1.1  +  0.1  ⋅  0.9  =  1.19

daniellex0x0xto · Accepted Answer

Let             x      ^        (  t  )  =      [                                        x            1                    (          t          )                                                  x            2                    (          t          )                      ]  . Then             x      ^       satisfies                    x        ^              ′    (  t  )  =  f  (  t  ,            x      ^        (  t  )  )for an appropriate function   f  :            R        2    →            R        2  . Now use Euler&#039;s method to estimate             x      ^        (  .1  ) and             x      ^        (  .2  ).For example,            x      ^        (  .1  )  ≈            x      ^        (  0  )  +  .1  f  (  0  ,            x      ^        (  0  )  )  .

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