Let a function f(x) be such that f "(x) = f ' (x) + ex and f(0) = 0 and f '(0) = 1, then find the value of ln \textrm{ } \left(\right. \frac{\left(\left(\right. \textrm{ } f \left(\right. 2 \left.\right) \textrm{ } \left.\right)\right)^{\textrm{ } 2}}{4} \left.\right).

Engineering

Mathematics

Determination of Function

Question

Let a function f(x) be such that f "(x) = f ' (x) + e^x and f(0) = 0 and f '(0) = 1, then find the value of $\ln (\frac{{(f (2))}^{2}}{4})$ .

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f "(x) – f ' (x) = e^x
⇒   e^–x · f '' (x) – e^–x · f ' (x) = 1
⇒ $\frac{d}{dx} (e^{- x} f' (x)) = 1 \Rightarrow \frac{f' (x)}{e^{x}} = x + C$
⇒   f '(x) = e^x (x + C) $∵$    (f(0) = 1 ⇒ C = 1)
⇒   f '(x) = e^x (x + 1)
⇒   f(x) = xe^x + k    $∵$ (f(0) = 0 ⇒ k = 0)
⇒   f(x) = xe^x
⇒ $\ln (\frac{{(f (2))}^{2}}{4})$ = 4.