Question

Let $f:R\to R$ be a continuous function satisfying
$f\left(x\right)+{\int }_0^{x}tf\left(t\right)dt+x^2=0$
for all $x\in R$. Then

• A. ${lim}_{x\to \infty }f\left(x\right)=2$
• B. ${lim}_{x\to \infty }f\left(x\right)=-2$
• C. f(x) has more than one point in common with the x-axis
• D. f(x) has no point in common with the x-axis

Answer 1

Answer: (B)

${lim}_{x\to \infty }f\left(x\right)=-2$

$f^{\prime }\left(x\right)+xf\left(x\right)+2x=0$

$\frac{dy}{dx}+xy=-2x$

$I.F.=e^{x^2/2}$

$\Rightarrow y.e^{x^2/2}=\int e^{x^2/2+C}$

$\Rightarrow e^{x^2/2}.y=-2e^{x^2/2}+C$

$\Rightarrow y=-2+C{e}^{x^2/2}$

$f\left(0\right)=0\Rightarrow C=2$

$f\left(x\right)=2[e^{-x^2/2}-1]$