Question

x(e^y)dy+((x^2)+1dx)/y=0

Answer 1

Let us solve the differential equation $xe^ydy+\frac{(x^2+1)dx}{y}=0,$ which is equivalent to $ye^ydy+\frac{(x^2+1)dx}{x}=0.$ It follows that $\int ye^ydy+\int\frac{(x^2+1)dx}{x}=C.$

Let $u=y, dv=e^ydy.$ Then $du=dy, v=e^y.$

Consequently, we get

$ye^y-\int e^ydy+\int (x+\frac{1}{x})dx=C.$

We conclude that the general solution of the differential equation is of the form:

$ye^y-e^y+\frac{x^2}{2}+\ln|x|=C.$