First slide

Methods of solving first order first degree differential equations

Question

The general solution of the differential equation $\frac{d y}{d x} + \sin \frac{x + y}{2} = \sin \frac{x - y}{2}$ is

Moderate

A

$logtan (\frac{y}{2}) = c - 2 \sin x$
B

$logtan (\frac{y}{4}) = c - 2 \sin \frac{x}{2}$
C

$logtan (\frac{y}{2} + \frac{π}{4}) = c - 2 \sin x$
D

$\log (\tan (\frac{y}{2} + \frac{π}{4})) = c - 2 \sin \frac{x}{2}$

Solution

$\begin{array}{l} \frac{d y}{d x} = \sin (\frac{x - y}{2}) - \sin (\frac{x + y}{2}) \\ \sin (A - B) - \sin (A + B) = - 2 \cos A \sin B \\ = - 2 \cos \frac{x}{2} \sin \frac{y}{2} \\ \Rightarrow \int \cos e c \frac{y}{2} \cdot d y = - 2 \int \cos \frac{x}{2} \cdot d x \\ \Rightarrow \frac{\ln |\tan \frac{y}{4}|}{\frac{1}{2}} = - 2 \cdot \frac{\sin (\frac{x}{2})}{\frac{1}{2}} + 2 c \\ \Rightarrow \ln | \tan (\frac{y}{4}) | = c - 2 \sin \frac{x}{2} \end{array}$

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