I have already posted question where I was asking about sketching Euler method.The explicit Euler method for numerically solving the begining values of differential equation x ′ = f ( t , x ) , x ( t 0 ) = x 0 on the interval I = [ t 0 , T ] is given by x k + 1 = x k + h f ( t k , x k ) , k = 0 , … , N − 1 with h = ( T − t 0 ) / N , N ∈ N . X k is an approximation of the exact solution x ( t ) of the begining values at time t k := t 0 + k h , k = 0 , . . . , N . By linear interpolation between the points ( t k , x k ) and ( t k + 1 , x k + 1 ) , k = 0 , . . . , N − 1 , we obtain a approximation solution x h ( t ).I need to calculate an approximation of the solution of the begining values at the point t = 1 x ′ = − t / x , x ( 0 ) = 1I need to use h = 0.5. Specify x h (1) and calculate the error, that is difference x h (1)− x(1), where x(1) is the value of the exact solution.I really don't know how to start. How can I calulate x or x h at all?

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I have already posted question where I was asking about sketching Euler method.The explicit Euler method for numerically solving the begining values of differential equation   x      ′    =  f  (  t  ,  x  )  ,  x  (      t    0    )  =      x    0   on the interval   I  =  [      t    0    ,  T  ] is given by      x          k      +      1        =      x    k    +  h  f  (      t    k    ,      x    k    )  ,  k  =  0  ,  …  ,  N  −  1 with   h  =  (  T  −      t    0    )      /    N  ,  N  ∈  N  .      X    k   is an approximation of the exact solution   x  (  t  ) of the begining values at time       t    k    :=      t    0    +  k  h  ,  k  =  0  ,  .  .  .  ,  N  . By linear interpolation between the points   (      t    k    ,      x    k    ) and   (      t          k      +      1        ,      x          k      +      1        )  ,  k  =  0  ,  .  .  .  ,  N  −  1  , we obtain a approximation solution       x    h    (  t  ).I need to calculate an approximation of the solution of the begining values at the point   t  =  1      x    ′    =  −  t      /    x  ,  x  (  0  )  =  1I need to use h = 0.5. Specify       x    h  (1) and calculate the error, that is difference       x    h  (1)−  x(1), where   x(1) is the value of the exact solution.I really don&#039;t know how to start. How can I calulate   x or       x    h   at all?

Deacon Nelson · Accepted Answer

You find   x by solving the differential equation. It has separeted variables, and the solutiion is easily found to be   x  (  t  )  =      1    −          t      2      . Then   x  (  1  )  =  0.To apply Euler&#039;s method we have       t    0    =  0,   T  =  1,       x    0    =  1,   h  =  0.5 and hence   N  =  2. Then      x          k      +      1        =      x    k    −  h              k            h              x      k        .Find       x    1   and then       x    2  .

I have already posted question where I was asking about sketching Euler method. The explicit Euler method for numerically solving the begining values of differential equation x'=f(t,x),x(t_0)=x_0 on the interval I=[t_0,T] is given by x_(k+1)=x_k+hf(t_k,x_k),k=0,…,N−1 with h=(T−t_0)/N,N in N.

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