Solve xdy=(y+xcos^2y/x)dx

$$x d y = ( y + x \cos ^ { 2 } \frac { y } { x } ) d x$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&x\neq 0\\d\in \mathrm{R}\text{, }&\exists n_{1}\in \mathrm{Z}\text{ : }x=\frac{2y}{2\pi n_{1}+\pi }\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y+x\left(\cos(\frac{y}{x})\right)^{2} by d.

$$xdy=\left(yd+x\left(\cos(\frac{y}{x})\right)^{2}d\right)x$$

Use the distributive property to multiply yd+x\left(\cos(\frac{y}{x})\right)^{2}d by x.

$$xdy=ydx+\left(\cos(\frac{y}{x})\right)^{2}dx^{2}$$

Subtract ydx from both sides.

$$xdy-ydx=\left(\cos(\frac{y}{x})\right)^{2}dx^{2}$$

Combine xdy and -ydx to get 0.

$$0=\left(\cos(\frac{y}{x})\right)^{2}dx^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$\left(\cos(\frac{y}{x})\right)^{2}dx^{2}=0$$

The equation is in standard form.

$$\left(\cos(\frac{y}{x})\right)^{2}x^{2}d=0$$

Divide 0 by \left(\cos(yx^{-1})\right)^{2}x^{2}.

$$d=0$$