Solve (2ax+by)ydx+(a+2by)xdy=0

$$( 2 a x + b y ) y d x + ( a + 2 b y ) x d y = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{3by}{2x+1}\text{, }&x\neq -\frac{1}{2}\\a\in \mathrm{C}\text{, }&x=0\text{ or }d=0\text{ or }\left(b=0\text{ and }x=-\frac{1}{2}\right)\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2ax+by by y.

$$\left(2axy+by^{2}\right)dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axy+by^{2} by d.

$$\left(2axyd+by^{2}d\right)x+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axyd+by^{2}d by x.

$$2aydx^{2}+by^{2}dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply a+2by by x.

$$2aydx^{2}+by^{2}dx+\left(ax+2byx\right)dy=0$$

Use the distributive property to multiply ax+2byx by d.

$$2aydx^{2}+by^{2}dx+\left(axd+2byxd\right)y=0$$

Use the distributive property to multiply axd+2byxd by y.

$$2aydx^{2}+by^{2}dx+axdy+2bxdy^{2}=0$$

Combine by^{2}dx and 2bxdy^{2} to get 3by^{2}dx.

$$2aydx^{2}+3by^{2}dx+axdy=0$$

Subtract 3by^{2}dx from both sides. Anything subtracted from zero gives its negation.

$$2aydx^{2}+axdy=-3by^{2}dx$$

Combine all terms containing a.

$$\left(2ydx^{2}+xdy\right)a=-3by^{2}dx$$

The equation is in standard form.

$$\left(dxy+2dyx^{2}\right)a=-3bdxy^{2}$$

Divide both sides by 2ydx^{2}+xdy.

$$\frac{\left(dxy+2dyx^{2}\right)a}{dxy+2dyx^{2}}=-\frac{3bdxy^{2}}{dxy+2dyx^{2}}$$

Dividing by 2ydx^{2}+xdy undoes the multiplication by 2ydx^{2}+xdy.

$$a=-\frac{3bdxy^{2}}{dxy+2dyx^{2}}$$

Divide -3by^{2}dx by 2ydx^{2}+xdy.

$$a=-\frac{3by}{2x+1}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{a\left(2x+1\right)}{3y}\text{, }&y\neq 0\\b\in \mathrm{C}\text{, }&x=0\text{ or }d=0\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2ax+by by y.

$$\left(2axy+by^{2}\right)dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axy+by^{2} by d.

$$\left(2axyd+by^{2}d\right)x+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axyd+by^{2}d by x.

$$2aydx^{2}+by^{2}dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply a+2by by x.

$$2aydx^{2}+by^{2}dx+\left(ax+2byx\right)dy=0$$

Use the distributive property to multiply ax+2byx by d.

$$2aydx^{2}+by^{2}dx+\left(axd+2byxd\right)y=0$$

Use the distributive property to multiply axd+2byxd by y.

$$2aydx^{2}+by^{2}dx+axdy+2bxdy^{2}=0$$

Combine by^{2}dx and 2bxdy^{2} to get 3by^{2}dx.

$$2aydx^{2}+3by^{2}dx+axdy=0$$

Subtract 2aydx^{2} from both sides. Anything subtracted from zero gives its negation.

$$3by^{2}dx+axdy=-2aydx^{2}$$

Subtract axdy from both sides.

$$3by^{2}dx=-2aydx^{2}-axdy$$

Reorder the terms.

$$3bdxy^{2}=-adxy-2adyx^{2}$$

The equation is in standard form.

$$3dxy^{2}b=-adxy-2adyx^{2}$$

Divide both sides by 3dxy^{2}.

$$\frac{3dxy^{2}b}{3dxy^{2}}=-\frac{adxy\left(2x+1\right)}{3dxy^{2}}$$

Dividing by 3dxy^{2} undoes the multiplication by 3dxy^{2}.

$$b=-\frac{adxy\left(2x+1\right)}{3dxy^{2}}$$

Divide -axyd\left(1+2x\right) by 3dxy^{2}.

$$b=-\frac{a\left(2x+1\right)}{3y}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{3by}{2x+1}\text{, }&x\neq -\frac{1}{2}\\a\in \mathrm{R}\text{, }&x=0\text{ or }d=0\text{ or }\left(b=0\text{ and }x=-\frac{1}{2}\right)\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2ax+by by y.

$$\left(2axy+by^{2}\right)dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axy+by^{2} by d.

$$\left(2axyd+by^{2}d\right)x+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axyd+by^{2}d by x.

$$2aydx^{2}+by^{2}dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply a+2by by x.

$$2aydx^{2}+by^{2}dx+\left(ax+2byx\right)dy=0$$

Use the distributive property to multiply ax+2byx by d.

$$2aydx^{2}+by^{2}dx+\left(axd+2byxd\right)y=0$$

Use the distributive property to multiply axd+2byxd by y.

$$2aydx^{2}+by^{2}dx+axdy+2bxdy^{2}=0$$

Combine by^{2}dx and 2bxdy^{2} to get 3by^{2}dx.

$$2aydx^{2}+3by^{2}dx+axdy=0$$

Subtract 3by^{2}dx from both sides. Anything subtracted from zero gives its negation.

$$2aydx^{2}+axdy=-3by^{2}dx$$

Combine all terms containing a.

$$\left(2ydx^{2}+xdy\right)a=-3by^{2}dx$$

The equation is in standard form.

$$\left(dxy+2dyx^{2}\right)a=-3bdxy^{2}$$

Divide both sides by 2ydx^{2}+xdy.

$$\frac{\left(dxy+2dyx^{2}\right)a}{dxy+2dyx^{2}}=-\frac{3bdxy^{2}}{dxy+2dyx^{2}}$$

Dividing by 2ydx^{2}+xdy undoes the multiplication by 2ydx^{2}+xdy.

$$a=-\frac{3bdxy^{2}}{dxy+2dyx^{2}}$$

Divide -3by^{2}dx by 2ydx^{2}+xdy.

$$a=-\frac{3by}{2x+1}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{a\left(2x+1\right)}{3y}\text{, }&y\neq 0\\b\in \mathrm{R}\text{, }&x=0\text{ or }d=0\text{ or }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 2ax+by by y.

$$\left(2axy+by^{2}\right)dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axy+by^{2} by d.

$$\left(2axyd+by^{2}d\right)x+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply 2axyd+by^{2}d by x.

$$2aydx^{2}+by^{2}dx+\left(a+2by\right)xdy=0$$

Use the distributive property to multiply a+2by by x.

$$2aydx^{2}+by^{2}dx+\left(ax+2byx\right)dy=0$$

Use the distributive property to multiply ax+2byx by d.

$$2aydx^{2}+by^{2}dx+\left(axd+2byxd\right)y=0$$

Use the distributive property to multiply axd+2byxd by y.

$$2aydx^{2}+by^{2}dx+axdy+2bxdy^{2}=0$$

Combine by^{2}dx and 2bxdy^{2} to get 3by^{2}dx.

$$2aydx^{2}+3by^{2}dx+axdy=0$$

Subtract 2aydx^{2} from both sides. Anything subtracted from zero gives its negation.

$$3by^{2}dx+axdy=-2aydx^{2}$$

Subtract axdy from both sides.

$$3by^{2}dx=-2aydx^{2}-axdy$$

Reorder the terms.

$$3bdxy^{2}=-adxy-2adyx^{2}$$

The equation is in standard form.

$$3dxy^{2}b=-adxy-2adyx^{2}$$

Divide both sides by 3dxy^{2}.

$$\frac{3dxy^{2}b}{3dxy^{2}}=-\frac{adxy\left(2x+1\right)}{3dxy^{2}}$$

Dividing by 3dxy^{2} undoes the multiplication by 3dxy^{2}.

$$b=-\frac{adxy\left(2x+1\right)}{3dxy^{2}}$$

Divide -axyd\left(1+2x\right) by 3dxy^{2}.

$$b=-\frac{a\left(2x+1\right)}{3y}$$