Question

Let y : ℝ → ℝ be the solution to the differential equation
\(\frac{d^2y}{dx^2}+2\frac{dy}{dx}+5y=1\)
satisfying y(0) = 0 and y'(0) = 1.
Then, \(\lim\limits_{x \rightarrow \infin}y(x)\) equals __________ (rounded off to two decimal places).

Answer 1

Correct Answer: 0.19

To solve this differential equation, we first find the complementary solution \( y_c \) and the particular solution \( y_p \). After solving, we apply the initial conditions \( y(0) = 0 \) and \( y'(0) = 1 \) to determine the constants. Finally, we compute the limit of \( y(x) \) as \( x \to \infty \), which yields the result approximately equal to 0.19.