(a) Find the particular solution of the differential equation x (d y)/(d x)+y+1/(1+x^2)=0, given that y(1)=0.

Given differential equation can be written as \(\displaystyle \frac{d y}{d x}+\frac{1}{x} y=\frac{-1}{x\left(1+x^{2}\right)}\) I.F. \(\displaystyle =e^{\int \frac{1}{x} d x}=e^{\log x}=x\)

Solution is \(\displaystyle y \cdot x=\int \frac{-1}{1+x^{2}} d x+C \Rightarrow x y=-\tan ^{-1} x+C\)

Now \(\displaystyle y(1)=0 \Rightarrow C=\frac{\pi}{4}\) ∴ Particular solution is \(\displaystyle x y=\frac{\pi}{4}-\tan ^{-1} x\)