First slide

Methods of solving first order first degree differential equations

Question

The solution of the differential equation $d y = \tan^{2} (x + y) d x$

Moderate

A

$\sin (2 x + 2 y) = 2 x - 2 y + c$
B

$\cos (2 x + 2 y) = 2 x - 2 y + c$
C

$\cos (2 x - 2 y) = 2 x + 2 y + c$
D

$\sin (2 x - 2 y) = 2 x + 2 y + c$

Solution

$\begin{array}{l} \frac{d y}{d x} = \tan^{2} (x + y) \\ Put x + y = t \Rightarrow 1 + \frac{d y}{d x} = \frac{d t}{d x} \\ \frac{d t}{d x} - 1 = \tan^{2} t \\ \Rightarrow \frac{d t}{d x} = 1 + \tan^{2} t = \sec^{2} t \\ \Rightarrow \cos^{2} t d t = d x \\ \Rightarrow \int \frac{1 + \cos 2 t}{2} d t = \int d x \\ \Rightarrow \frac{1}{2} (t + \frac{\sin 2 t}{2}) = x + c \end{array}$
$\begin{array}{l} \Rightarrow \frac{1}{4} [2 (x + y) + \sin 2 (x + y)] = x + c \\ \Rightarrow \sin (2 x + 2 y) = 2 x - 2 y + c \end{array}$

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