DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

$(\sin\text{x}+\cos\text{x})\text{dy}+(\cos\text{x}+\sin\text{x})\text{dx}=0$

$\Rightarrow{\text{dy}}=-\Big(\frac{\cos\text{x}-\sin\text{x}}{\sin\text{x}-\cos\text{x}}\Big)\text{dx}$

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.