Let be a non-zero real number. Suppose is a differentiable function such that f(0) = 2 and . If f'(x) = f(x) + 3, for all , then is equal to ________. [2024]

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Solution

Q.
Let $α$ be a non-zero real number. Suppose $f : R \to R$ is a differentiable function such that f(0) = 2 and $\lim_{x \to - \infty} f (x) = 1$ . If f'(x) = $α$ f(x) + 3, for all $x \in R$ , then $f (- \log_{e} 2)$ is equal to ________. [2024]

1 7

2 9

3 3

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Ans.
*

We have, $f' (x) = α f (x) + 3 \Rightarrow f' (x) - α f (x) = 3$

Now, Let $α > 0, IF = e^{- α \int d x} = e^{- α x}$

$∴$ Solution is given by, $e^{- α x} f (x) = 3 \int e^{- α x} d x + C$

$\Rightarrow f (x) = \frac{- 3}{α} + C e^{α x}$

Now, let $α > 0$ , we have $\lim_{x \to - \infty} f (x) = 1$

$\Rightarrow \frac{- 3}{α} + C \lim_{x \to - \infty} e^{α x} = 1$

Let $α < 0, \lim_{x \to \infty} f (x) = - \frac{3}{α} + C \lim_{x \to - \infty} e^{α x} = 1$

$\Rightarrow \frac{- 3}{α} + \infty = 1$ which is not possible

$∴$ Value of $α$ does not exist.

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