Find a polynomial f(x) that satisfies this equation: xf " (x) + f(x) = x^2

Karsyn Beltran 2022-08-01 Answered
Find a polynomial f(x) that satisfies this equation:
x f ( x ) + f ( x ) = x 2
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Answers (1)

esbalatzaj
Answered 2022-08-02 Author has 15 answers
Look for a solution of the form:
f ( x ) = r = 0 n a r x r
For some set of constants a r
Then f ( x ) = r = 1 n a r r x r 1
f ( x ) = r = 2 n a r r ( r 1 ) x r 2
So we have
x r = 2 n a r r ( r 1 ) x r 2 + r = 0 n a r x r = x 2
Equating like terms on lhs and rhs we see that:
6 a 3 + a 2 = 1 (for the x 2 terms)
2 a 2 + a 1 = 0 (for the x terms)
a 0 = 0 (for the constant terms)
But we must have a 2 = 1 for the squared termsbecause there's just x 2 on rhs
Hence a 1 = 2
a 3 = 0
So f ( x ) = x 2 2 x
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