Question

For a ∈ ℝ, let ya(x) be the solution of the differential equation
$\frac{dy}{dx}+2y=\frac{1}{1+x^2}$ for x ∈ $\R$
satisfying y(0) = a. Then, which one of the following is TRUE ?

• A. $\lim\limits_{x\rightarrow \infin}y_a(x)=0$ for every a ∈ $\R$
• B. $\lim\limits_{x\rightarrow \infin}y_a(x)=1$ for every a ∈ $\R$
• C. There exists an a ∈ ℝ such that $\lim\limits_{x \rightarrow \infin}y_a(x)$ exists but its value is different from 0 and 1
• D. There exists an a ∈ ℝ for which $\lim \limits_{x \rightarrow \infin}y_a(x)$ does not exist

Answer 1

Answer: (A)

$\lim\limits_{x\rightarrow \infin}y_a(x)=0$ for every a ∈ $\R$

Answer 2

The correct option is (A) : $\lim\limits_{x\rightarrow \infin}y_a(x)=0$ for every a ∈ $\R$.