Question

Let $f$ be a function satisfying the equation $f\left(x\right)+3\underset{-1}{\overset{1}{\int }}(xy-x^2{y}^2)f\left(y\right)dy=x^3.$ Then the value of $f\left(5\right)$ is

Answer 1

$f\left(x\right)+3\underset{-1}{\overset{1}{\int }}(xy-x^2{y}^2)f\left(y\right)dy=x^3$
$\Rightarrow f\left(x\right)=x^3+3x^2\underset{-1}{\overset{1}{\int }}{y}^{2}f\left(y\right)dy-3x\underset{-1}{\overset{1}{\int }}yf\left(y\right)dy$
$\Rightarrow f\left(x\right)=x^3+3x^2\alpha -3x\beta ,$
where $\alpha =\underset{-1}{\overset{1}{\int }}{y}^{2}f\left(y\right)dy$
$\Rightarrow \alpha =\underset{-1}{\overset{1}{\int }}(y^5+3y^4\alpha -3y^3\beta )dy$
$\Rightarrow \alpha =0$
and $\beta =\underset{-1}{\overset{1}{\int }}yf\left(y\right)dy$
$\Rightarrow \beta =\underset{-1}{\overset{1}{\int }}(y^4+3y^3\alpha -3y^2\beta )dy$
$\Rightarrow \beta =\frac{2}{5}-2\beta$
$\Rightarrow \beta =\frac{2}{15}$
$\therefore f\left(x\right)=x^3-\frac{2}{5}x$
Hence, $f\left(5\right)=123$