Find-a-particular-solution-of-the-equation-f-x-f-x-such-that-f-x-2-for-x-2-

Question Number 130893 by EDWIN88 last updated on 30/Jan/21

F i n d a p a r t i c u l a r s o l u t i o n

o f t h e e q u a t i o n f^{'} (x) = f (x)

s u c h t h a t f (x) = 2 f o r x = 2

Answered by mr W last updated on 30/Jan/21

y^{'} = y

\frac{d y}{y} = d x

\ln y = x + C

\Rightarrow y = {c e}^{x}

\Rightarrow f (2) = {c e}^{2} = 2 \Rightarrow c = 2 e^{- 2}

\Rightarrow f (x) = y = 2 e^{x - 2}

Answered by benjo_mathlover last updated on 30/Jan/21

y^{'} - y = 0 \Rightarrow λ - 1 = 0; λ = 1

y (x) = C e^{x} \Rightarrow y (2) = {Ce}^{2} = 2; C = {2 e}^{- 2}

∴ y (x) = {2 e}^{- 2} . e^{x} = {2 e}^{x - 2}