Solve the following differential equation ( + )dy+( + )dx=0 - Vidyadip

Question

Solve the following differential equation
$(\sin\text{x}+\cos\text{x})\text{dy}+(\cos\text{x}+\sin\text{x})\text{dx}=0$

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Answer

We have,$(\sin\text{x}+\cos\text{x})\text{dy}+(\cos\text{x}+\sin\text{x})\text{dx}=0$
$\Rightarrow{\text{dy}}=-\Big(\frac{\cos\text{x}-\sin\text{x}}{\sin\text{x}-\cos\text{x}}\Big)\text{dx}$
Integrating both sides, we get
$\int{\text{dy}}=-\int\Big(\frac{\cos\text{x}-\sin\text{x}}{\sin\text{x}-\cos\text{x}}\Big)\text{dx}$
$\Rightarrow\text{y}=-\int\Big(\frac{\cos\text{x}-\sin\text{x}}{\sin\text{x}-\cos\text{x}}\Big)\text{dx}$
Putting $\sin\text{x}+\cos\text{x}=\text{t}$
$\Rightarrow(\cos\text{x}-\sin\text{x})\text{dx}=\text{dt}$
$\therefore\text{y}=-\int\frac{\text{dt}}{\text{t}}$
$\Rightarrow\text{y}=-\log|\text{t}|+\text{C}$
$\Rightarrow\text{y}=-\log|\sin\text{x}+\cos\text{x}|+\text{C}$
$\Rightarrow\text{y}=+\log|\sin\text{x}+\cos\text{x}|=\text{C}$
Hence, $\text{y}=+\log|\sin\text{x}+\cos\text{x}|=\text{C}$ is the solution to the given differential equation.

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