ONE MORE STEP TOWARD BETTER TOMORROW : CHAPTER 15

Tuesday, 5 August 2014

CHAPTER 15 - Miscellaneous Examples

Example: 1

Let $f:{\mathbb{R}^ + } \to \mathbb{R}$ be a differentiable function such that

$f\left( x \right) = e - \left( {x - 1} \right)\ln \dfrac{x}{e} + \int\limits_1^x {f\left( x \right)dx}$

Find a simple expression for $f(x)$ .

Solution: 1

Step-1

Differentiating the given relation, we have

	$\dfrac{{df}}{{dx}} = - \dfrac{{\left( {x - 1} \right)}}{x} - \ln \dfrac{x}{e} + f$
	$\Rightarrow \dfrac{{df}}{{dx}} - f = \dfrac{1}{x} - \ln x$

Step-2

This is evidently a first-order linear DE; the IF is ${e^{\int { - dx} }} = {e^{ - x}}$ . Multiplying it across both sides of the DE renders the DE exact and its solution is given by

	${e^{ - x}} \cdot f = \int {{e^{ - x}}\left( {\dfrac{1}{x} - \ln x} \right)dx}$
	$= {e^{ - x}}\ln x + C$
	$\Rightarrow f\left( x \right) = \,\ln x + C\,{e^x}$	$\ldots (1)$