The form f ( x ) y n </msup> ( x ) + . . . u

Camille Flynn

Camille Flynn

Answered question

2022-05-19

The form f ( x ) y n ( x ) + . . . u ( x ) y ( x ) = h ( x ) is supposed to be the definition of lineaity in diff equations. It excludes functions of y in the right hand side, but is multiplication of y by another function allowed in the right hand side allowed? It seems to be the case sometimes but not all of the time, as only composition of linear functions gives linear functions. Can't we just move it to the other side with the other y and call that form linear as well?

Answer & Explanation

kovilovop2

kovilovop2

Beginner2022-05-20Added 7 answers

One can always move terms from one side of an equation to the other by subtraction.
n ( x ) y ( n ) ( x ) + n 1 ( x ) y ( n 1 ) ( x ) + + 1 ( x ) y ( x ) + 0 ( h ) y ( x ) + 1 ( x ) = r n ( x ) y ( n ) ( x ) + r n 1 ( x ) y ( n 1 ) ( x ) + + r 1 ( x ) y ( x ) + r 0 ( h ) y ( x ) + r 1 ( x )
if and only if
[ n ( x ) r n ( x ) ] y ( n ) ( x ) + [ n 1 ( x ) r n 1 ( x ) ] y ( n 1 ) ( x ) + + [ 1 ( x ) r 1 ( x ) ] y ( x ) + [ 0 ( x ) r 0 ( x ) ] y ( x ) = r 1 ( x ) 1 ( x ) .

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