First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation $(\sin x + \cos x) d y + (\cos x - \sin x) d x = 0$ is

Moderate

A

$e^{x} (\sin x + \cos x) + c = 0$
B

$e^{y} (\sin x + \cos x) = c$
C

$e^{x} (\cos x - \sin x) = c$
D

$e^{y} (\sin x - \cos x) = c$

Solution

$\begin{array}{l} (\sin x + \cos x) d y + (\cos x - \sin x) d x = 0 \\ \int d y = \int \frac{\sin x - \cos x}{\sin x + \cos x} d x \\ Put \cos x + \sin x = t \Rightarrow y = - \int \frac{d t}{t} = - \ln t + \ln c \\ \Rightarrow y = \ln (\frac{c}{\sin x + \cos x}) \\ \Rightarrow e^{y} = \frac{c}{\sin x + \cos x} \Rightarrow e^{y} (\sin x + \cos x) = c \end{array}$

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