Question

Solution of differential equation $\left(x\cos x-\sin x\right)dx=\frac{x}{y}\sin xdy$ is

• A. $\sin x=\ln \left|xy\right|+c$
• B. $\ln \left|\frac{\sin x}{x}\right|=y+c$
• C. $\left|\frac{\sin x}{xy}\right|=c$
• D. None of these

Answer 1

Answer: (C)

$\left|\frac{\sin x}{xy}\right|=c$

$(x\cos x-\sin x)dx=\frac{x}{y}\sin xdy$

$\Rightarrow \frac{(x\cos x-\sin x)}{x\sin x}dx=\frac{1}{y}dy$

$\Rightarrow (\cot x-\frac{1}{x})dx=\frac{1}{y}dy$

$\Rightarrow \log |\sin x|-\log |x|=\log |y|+c_1$

$\Rightarrow \log \left|\frac{\sin x}{xy}\right|=C_1$

$\Rightarrow \left|\frac{\sin x}{xy}\right|=c[e^{c_1}=c]$