Solution of differential equation x^ 2 =1+((x)/(y))^ -1 (d y)/(d...

x^{2}=e^{\left(\frac{x}{y}\right)^{-1}\left(\frac{d y}{d x}\right)} \Rightarrow x^{2}=e^{\left(\frac{y}{x}\right)\left(\frac{d y}{d x}\right)}\Rightarrow \ln x^{2}=\frac{y}{x} \frac{d y}{d x}$or$\int x \ln x^{2} d x=\int y d y$Put$x^{2}=t \Rightarrow 2 x d x=d t \therefore \frac{1}{2} \int \ln t d t=\frac{y^{2}}{2}C+t \ln t-t=y^{2}$or$y^{2}=x^{2}\left(\ln x^{2}-1\right)+C