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

The Correct Option is A

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