Question

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

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

Answer 1

Answer: (D)

$ \frac{d y}{d x}+x y=x \Rightarrow y \cdot e^{\frac{x^2}{2}}=\int x \cdot e^{\frac{x^2}{2}} d x+c$
$y \cdot e^{\frac{x^2}{2}}=e^{\frac{x^2}{2}}+c, y(0)=2 \Rightarrow c=1$
$\Rightarrow y=1+e^{\frac{-x^2}{2}}$
$\Rightarrow \underset{x \rightarrow 0}{\text{Lim}} \frac{e^{\frac{-x^2}{2}}-1}{\frac{-x^2}{2}}\left(\frac{-1}{2}\right)=\frac{-1}{2}$