289_pdfsam_math 54 differential equation solutions odd

289_pdfsam_math 54 differential equation solutions odd -...

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Exercises 5.3 x 1; n +1 = x 1; n + h 2 [ f 1 ( t n ,x 1; n 2; n ) + f 1 ( t n + h, x 1; n + hf 1 ( t n 1; n 2; n ) 2; n + hf 2 ( t n 1; n 2; n ))] , x 2; n +1 = x 2; n + h 2 [ f 2 ( t n 1; n 2; n ) + f 2 ( t n + h, x 1; n + hf 1 ( t n 1; n 2; n ) 2; n + hf 2 ( t n 1; n 2; n ))] . The approach can be used more generally for systems of m equations in normal form. Suppose we want to approximate the solution x 1 ( t ), x 2 ( t ), ... , x m ( t ) to the system x 0 1 = f 1 ( t, x 1 2 ,...,x m ) , x 0 2 = f 2 ( t, x 1 2 m ) , . . . x 0 m = f m ( t, x 1 2 m ) , with the initial conditions x 1 ( t 0 )= a 1 2 ( t 0 a 2 , ... , x m ( t 0 a m . We adapt the recursive formulas above to obtain t n +1 = t n + h, n =0 , 1 , 2 ,... ; x 1; n +1 = x 1; n + h 2 [ f 1 ( t n 1; n 2; n m ; n )+ f 1 ( t n + h, x 1; n + hf 1 ( t n 1; n 2; n m ; n ) , x 2; n + hf 2 ( t n 1; n 2; n m ; n ) m ; n + hf m ( t n 1; n 2; n m ; n ))] , x 2; n +1 = x 2; n + h 2 [ f 2 ( t n 1; n 2; n m ; n f 2 ( t n + h, x 1; n + hf 1 ( t n 1; n 2; n m ; n ) , x 2; n + hf 2 ( t n 1; n 2; n m ; n ) m ; n + hf m ( t n 1; n 2; n m ; n ))] , . . . x m ; n +1 = x m ; n + h 2 [ f m ( t n 1; n 2; n m ; n f m ( t n + h, x 1; n + hf 1 ( t n 1; n 2; n m ; n ) , x 2; n + hf 2 ( t n 1; n 2; n m ; n ) m ;
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