Question

Consider the differential equation \[ \frac{dy}{dx} + 10y = f(x), x > 0, \] where \( f(x) \) is a continuous function such that \( \lim_{x \to \infty} f(x) = 1. \) Then the value of \( \lim_{x \to \infty} y(x) \) is .................

Hint:

For first-order linear ODEs of the form \( y' + ay = f(x) \), if \( f(x) \to L \) as \( x \to \infty \), then \( \lim_{x \to \infty} y = \dfrac{L}{a}. \)

Answer 1

Correct Answer: 0.1

Step 1: Standard form and integrating factor.
The given equation is linear: \[ \frac{dy}{dx} + 10y = f(x). \] Integrating factor (I.F.) = \( e^{10x} \).

Step 2: Multiply both sides by I.F.
\[ \frac{d}{dx}(y e^{10x}) = f(x) e^{10x}. \] Integrate both sides: \[ y e^{10x} = \int f(x) e^{10x} dx + C. \]

Step 3: Take limit as \( x \to \infty \).
If \( f(x) \to 1 \), for large \( x \), \[ y \approx e^{-10x} \int e^{10x} dx = e^{-10x} \cdot \frac{1}{10} e^{10x} = \frac{1}{10}. \] Hence \( \lim_{x \to \infty} y = \frac{1}{10}. \)

Final Answer: \[ \boxed{\frac{1}{10}} \]