Question

Let $f:R\to R$ be a continuous function satisfying

$f\left(x\right)+{\int }_0^{x}tf\left(t\right)dt+x^2=0$
$forallxϵR.Then$

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

$f^{\prime }\left(x\right)+xf\left(x\right)+2x=0\frac{dy}{dx}+xy=-2xI.F.=e^{\frac{x^2}{2}}\Rightarrow y.e^{\frac{x^2}{2}}=\int e^{\frac{x^2}{2}}(-2x)dx+C{e}^{\frac{x^2}{2}}.y=-2e^{\frac{x^2}{2}}+Cy=-2+C{e}^{\frac{i{x}^2}{2}}f\left(0\right)=0\Rightarrow C=2f\left(x\right)=2\lfloor e^{\frac{-x^2}{2}}-1\rfloor$