First slide

Introduction to Differential equations

Question

$x \cos x \frac{d y}{d x} + y (x \sin x + \cos x) = 1 then xy =$

Easy

A

$\sin x + c . \cos x$
B

$\cos x + c \sin x$
C

$c . \sin x - \cos x$
D

$c . \cos x - \sin x$

Solution

$\begin{array}{l} \frac{d y}{d x} + (\frac{x \sin x + \cos x}{x \cos x}) y = \frac{1}{x \cos x} \\ I.F = \exp \int \frac{x \sin x + \cos x}{x \cos x} d x \end{array}$
$\begin{array}{l} \exp \int (\tan x + \frac{1}{x}) d x \\ = x \cdot \sec x \\ Solution is y \cdot x \sec x = \int \frac{x \cdot \sec x}{x \cos x} \\ = \tan x + c \\ ∴ x y = \sin x + c \cdot \cos x \end{array}$

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