Question

Let $y=f(x)$ be a particular solution of the differential equation $dy+xydx=xdx$ which satisfies $y(0)=2$, then $\underset{x\to 0}{\mathrm{Lim}}\frac{f(x)-2}{x^2}$ equals

• A. 1
• B. $\frac{1}{2}$
• C. -1
• D. $\frac{-1}{2}$

Answer 1

Answer: (D)

$\frac{dy}{dx}+xy=x\Rightarrow y\cdot e^{\frac{x^2}{2}}=\int x\cdot e^{\frac{x^2}{2}}dx+c$
$y\cdot e^{\frac{x^2}{2}}=e^{\frac{x^2}{2}}+c,y(0)=2\Rightarrow c=1$
$\Rightarrow y=1+e^{\frac{-x^2}{2}}$
$\Rightarrow \underset{x\to 0}{\mathrm{Lim}}\frac{e^{\frac{-x^2}{2}}-1}{\frac{-x^2}{2}}\left(\frac{-1}{2}\right)=\frac{-1}{2}$