First slide

Introduction to Differential equations

Question

$If f (1) = 0 and \frac{d f}{d x} > f (x) \forall x \geq 1, then$

Moderate

A

$f (x) > 0$
B

$f (x) > 1$
C

$f (x) > e$
D

$f (x) > \frac{1}{e}$

Solution

$\begin{array}{l} \frac{d f}{d x} - f (x) > 0 \Rightarrow e^{- x} (\frac{d f}{d x} - f (x)) > e \\ \Rightarrow \frac{d}{d x} (e^{- x} \cdot f (x)) > 0 \Rightarrow e^{- x} \cdot f (x) is on increasing function \\ f (1) = 0 \to e^{- x} \cdot f (x) > 0 \to f (x) > 0 \end{array}$

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