Solve dy/y=dx/4x-x^2

$$\frac { d y } { y } = \frac { d x } { 4 x - x ^ { 2 } }$$

Solve for d

$$\left\{\begin{matrix}d=0\text{, }&y\neq 0\text{ and }x\neq 4\text{ and }x\neq 0\\d\in \mathrm{R}\text{, }&y\neq 0\text{ and }x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by xy\left(x-4\right), the least common multiple of y,4x-x^{2}.

$$x\left(x-4\right)dy=-ydx$$

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

$$\left(x^{2}-4x\right)dy=-ydx$$

Use the distributive property to multiply x^{2}-4x by d.

$$\left(x^{2}d-4xd\right)y=-ydx$$

Use the distributive property to multiply x^{2}d-4xd by y.

$$x^{2}dy-4xdy=-ydx$$

Add ydx to both sides.

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

Combine -4xdy and ydx to get -3xdy.

$$x^{2}dy-3xdy=0$$

Combine all terms containing d.

$$\left(x^{2}y-3xy\right)d=0$$

The equation is in standard form.

$$\left(yx^{2}-3xy\right)d=0$$

Divide 0 by x^{2}y-3xy.

$$d=0$$

Solve for x

$$\left\{\begin{matrix}x=3\text{, }&y\neq 0\\x\in \mathrm{R}\setminus 4,0\text{, }&d=0\text{ and }y\neq 0\end{matrix}\right.$$