First slide

Methods of solving first order first degree differential equations

Question

$If x d y - y d x + x \cos (\ln x) d x = 0, y (1) = 1, then y (e)$

Moderate

A

$e (1 - \cos 1)$
B

$e (1 - \sin 1)$
C

$e (1 + \cos 1)$
D

$e (1 + \sin 1)$

Solution

$\begin{array}{l} x d y - y d x = - x \cos (\log x) d x \\ \frac{x d y - y d x}{x^{2}} = - \frac{\cos (\log x) d x}{x} \\ d (\frac{y}{x}) = - \frac{\cos (\log x) d x}{x} \\ By integrating \\ \frac{y}{x} = - \sin (\log x) + c \\ y = x (- \sin (\log x) + c) \\ y (1) = 1 \Rightarrow c = 1 \\ ∴ y (e) = e (- \sin (1) + 1) \end{array}$

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