Solve (2x-5y)dx+(4x-y)dy=0

$$( 2 x - 5 y ) d x + ( 4 x - y ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&x=-\frac{y}{2}\text{ or }x=y\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2x-5y by d.

$$\left(2xd-5yd\right)x+\left(4x-y\right)dy=0$$

Use the distributive property to multiply 2xd-5yd by x.

$$2dx^{2}-5ydx+\left(4x-y\right)dy=0$$

Use the distributive property to multiply 4x-y by d.

$$2dx^{2}-5ydx+\left(4xd-yd\right)y=0$$

Use the distributive property to multiply 4xd-yd by y.

$$2dx^{2}-5ydx+4xdy-dy^{2}=0$$

Combine -5ydx and 4xdy to get -ydx.

$$2dx^{2}-ydx-dy^{2}=0$$

Combine all terms containing d.

$$\left(2x^{2}-yx-y^{2}\right)d=0$$

The equation is in standard form.

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

Divide 0 by 2x^{2}-yx-y^{2}.

$$d=0$$

Solve for d

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

Steps for Solving Linear Equation

Use the distributive property to multiply 2x-5y by d.

$$\left(2xd-5yd\right)x+\left(4x-y\right)dy=0$$

Use the distributive property to multiply 2xd-5yd by x.

$$2dx^{2}-5ydx+\left(4x-y\right)dy=0$$

Use the distributive property to multiply 4x-y by d.

$$2dx^{2}-5ydx+\left(4xd-yd\right)y=0$$

Use the distributive property to multiply 4xd-yd by y.

$$2dx^{2}-5ydx+4xdy-dy^{2}=0$$

Combine -5ydx and 4xdy to get -ydx.

$$2dx^{2}-ydx-dy^{2}=0$$

Combine all terms containing d.

$$\left(2x^{2}-yx-y^{2}\right)d=0$$

The equation is in standard form.

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

Divide 0 by 2x^{2}-yx-y^{2}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-\frac{y}{2}\text{; }x=y\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=-\frac{y}{2}\text{; }x=y\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$