If lim x → −∞ f(x) = 1 and f'(x) = αf(x) + 3 and f : R → R with f(0) = 2 then |α| is equal to ______.

Question

If \(\lim\limits_{x \to - \infty} f(x) = 1\), and f'(x) = αf(x) + 3 and f : R → R with f(0) = 2 then |α| is equal to ______.

Answer 1

\(\frac{dy}{dx} = (αy + 3)\)

⇒ ln|αy + 3| = x + c

⇒ αy + 3 = ke^x

⇒ f(0) = 2

⇒ 2α + 3 = ke⁰

⇒ k = (2α + 3)

⇒ \(y = \frac{(2\alpha + 3)e^x - 3}{\alpha}\)

\(\lim\limits_{ x \to -\infty} \frac{(2\alpha + 3)e^x - 3}{\alpha} = \frac{-3}\alpha = 1\)

⇒ \(\alpha = -3\)