Question

Solution of the differential equation $ \frac{dy}{dx}+\frac{y}{x}=\sin x $ is:

• A. $ x(y+\cos x)=\sin x+c $
• B. $ x(y-\cos x)=\sin x+c $
• C. $ x(y\cos x)=\sin x+c $
• D. $ x(y-\cos x)=\cos x+c $
• E. $ x(y+\cos x)=\cos x+c $

Answer 1

Answer: (A)

$ \because $ $ \frac{dy}{dx}+\frac{y}{x}=\sin x $
Here, $ P=\frac{1}{x} $ and $ Q=\sin x $
$ \therefore $ $ IF={{e}^{\int{p}\,dx}}={{e}^{\int{1/x}\,dx}}=x $
$ \therefore $ Solution is $ y(IF)=\int{Q(IF)}\,dx+c $
$ \Rightarrow $ $ y.x=\int{x\,\sin x\,dx+c} $
$ \Rightarrow $ $ xy=-x\cos x+\sin x+c $
$ \Rightarrow $ $ x(y+\cos x)=\sin x+c $