Let ' f ' is a differentiable function such that f ′ ( x ) = f ( x ) + ∫ 0 2 f ( x ) d x , f ( 0 ) = 3 4 − e 2 then the value of [ f ( 2 )] where [.] denotes the greatest integer ≤ x is.

Question

Let ' f ' is a differentiable function such that f ′ ( x ) = f ( x ) + ∫ 0 2 ​ f ( x ) d x , f ( 0 ) = 3 4 − e 2 ​ then the value of [ f ( 2 )] where [.] denotes the greatest integer ≤ x is.

TestHub · Accepted Answer

Given f ′ ( x ) = f ( x ) + A where A = ∫ 0 2 ​ f ( x ) d x Solving - (1) f ( x ) = λ ( e x − 1 ) + 3 4 − e 2 ​ ∵ ∫ 0 2 ​ f ( x ) d x = A ⇒ λ = 1 and A = 3 e 2 − 1 ​ ​ ∴ f ( x ) = e x − 1 + 3 4 − e 2 ​ = e x − 3 1 ​ ( e 2 − 1 ) ​ f ( 2 ) = 3 2 e 2 + 1 ​ ∴ [ f ( 2 )] = 5 ​

Mathematics - Differential Equation Question with Solution | TestHub

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