Differential Equations homogenous same degree with $\sec(y/x)$

Question

How do you solve this equation? $$\left(x\sec\frac yx-y\right)dx + xdy=0$$ They are homogeneous of the same degree but I don't know if I should use that method or not...

Answer 1

Let $y=xt $ and so $dy=tdx+xdt$ therefore $$\left(x\sec\frac yx-y\right)dx + xdy=(\sec t-t)dx+(tdx+xdt)=0$$ so we have $$\sec tdx+xdt=0$$ also $$\frac{dx}{x}=-\frac{dt}{\sec t}=-\cos t dt$$ so that $$\ln|x|+C=-\cos t=-\cos(\frac{y}{x})$$