Given-f-x-0-1-x-y-2-f-y-dy-2x-2-3x-1-find-f-x-

Question Number 189066 by cortano12 last updated on 11/Mar/23

Given f (x) + \underset{0}{\int^{1}} {(x + y)}^{2} f (y) dy = {2 x}^{2} - 3 x + 1

find f (x) .

Answered by horsebrand11 last updated on 11/Mar/23

2 x^{2} - 3 x + 1 = f (x) + \underset{0}{\int^{1}} (x^{2} + 2 x y + y^{2}) d y

2 x^{2} - 3 x + 1 = f (x) + x^{2} \underset{0}{\int^{1}} f (y) d y + 2 x \underset{0}{\int^{1}} y f (y) d y + \underset{0}{\int^{1}} y^{2} f (y) d y

let {\begin{cases} p = \underset{0}{\int^{1}} f (y) d y \\ q = \underset{0}{\int^{1}} y f (y) d y \\ r = \underset{0}{\int^{1}} y^{2} f (y) d y \end{cases}

f (x) = 2 x^{2} - {p x}^{2} - 3 x - 2 q x + 1 - r

f (x) = (2 - p) x^{2} - (3 + 2 q) x + (1 - r)

Answered by mr W last updated on 11/Mar/23

f (x) + \int_{0}^{1} {(x + y)}^{2} f (y) d y = 2 x^{2} - 3 x + 1

f (x) + \int_{0}^{1} (x^{2} + 2 x y + y^{2}) f (y) d y = 2 x^{2} - 3 x + 1

f (x) + x^{2} \int_{0}^{1} f (y) d y + 2 x \int_{0}^{1} y f (y) d y + \int_{0}^{1} y^{2} f (y) d y = 2 x^{2} - 3 x + 1

f (x) + {A x}^{2} + 2 B x + C = 2 x^{2} - 3 x + 1

\Rightarrow f (x) = (2 - A) x^{2} - (3 + 2 B) x + (1 - C)

A = \int_{0}^{1} f (y) d y = \int_{0}^{1} [(2 - A) y^{2} - (3 + 2 B) y + (1 - C)] d y

A = (2 - A) \frac{1}{3} - (3 + 2 B) \frac{1}{2} + (1 - C)

\Rightarrow 8 A + 6 B + 6 C = 1 \dots (i)

B = \int_{0}^{1} y f (y) d y = \int_{0}^{1} [(2 - A) y^{3} - (3 + 2 B) y^{2} + (1 - C) y] d y

B = (2 - A) \frac{1}{4} - (3 + 2 B) \frac{1}{3} + (1 - C) \frac{1}{2}

\Rightarrow 3 A + 20 B + 6 C = 0 \dots (i i)

C = \int_{0}^{1} y^{2} f (y) d y = \int_{0}^{1} [(2 - A) y^{4} - (3 + 2 B) y^{3} + (1 - C) y^{2}] d y

C = (2 - A) \frac{1}{5} - (3 + 2 B) \frac{1}{4} + (1 - C) \frac{1}{3}

\Rightarrow 12 A + 30 B + 80 C = - 1 \dots (i i i)

f r o m (i), (i i), (i i i) :

A = \frac{71}{453}, B = - \frac{7}{453}, C = - \frac{356}{1359}

\Rightarrow f (x) = \frac{835 x^{2}}{453} - \frac{1345 x}{453} + \frac{1715}{1359}

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