Solve dy/dx=2x+y/x+ydy

$$\frac { d y } { d x } = \frac { 2 x + y } { x + y } d y$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq -2x\text{ and }y\neq 0\text{ and }y\neq -x\\d\in \mathrm{R}\text{, }&\left(y=-2x\text{ or }y=0\right)\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+y.

$$\left(x+y\right)\frac{\mathrm{d}(y)}{\mathrm{d}x}=\left(2x+y\right)dy$$

Use the distributive property to multiply x+y by \frac{\mathrm{d}(y)}{\mathrm{d}x}.

$$x\frac{\mathrm{d}(y)}{\mathrm{d}x}+y\frac{\mathrm{d}(y)}{\mathrm{d}x}=\left(2x+y\right)dy$$

Use the distributive property to multiply 2x+y by d.

$$x\frac{\mathrm{d}(y)}{\mathrm{d}x}+y\frac{\mathrm{d}(y)}{\mathrm{d}x}=\left(2xd+yd\right)y$$

Use the distributive property to multiply 2xd+yd by y.

$$x\frac{\mathrm{d}(y)}{\mathrm{d}x}+y\frac{\mathrm{d}(y)}{\mathrm{d}x}=2xdy+dy^{2}$$

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

$$2xdy+dy^{2}=x\frac{\mathrm{d}(y)}{\mathrm{d}x}+y\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

Combine all terms containing d.

$$\left(2xy+y^{2}\right)d=x\frac{\mathrm{d}(y)}{\mathrm{d}x}+y\frac{\mathrm{d}(y)}{\mathrm{d}x}$$

The equation is in standard form.

$$\left(2xy+y^{2}\right)d=0$$

Divide 0 by 2xy+y^{2}.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y}{2}\text{, }&d\neq 0\text{ and }y\neq 0\\x\neq -y\text{, }&d=0\text{ or }y=0\end{matrix}\right.$$