Solve y^prime-ay/x=x+1/x

$$y ^ { \prime } - a \frac { y } { x } = \frac { x + 1 } { x }$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{x+1}{y}\text{, }&y\neq 0\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&y=0\text{ and }x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$x\frac{\mathrm{d}}{\mathrm{d}x}(y)-ay=x+1$$

Subtract x\frac{\mathrm{d}}{\mathrm{d}x}(y) from both sides.

$$-ay=x+1-x\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Reorder the terms.

$$-ay=-x\frac{\mathrm{d}}{\mathrm{d}x}(y)+x+1$$

The equation is in standard form.

$$\left(-y\right)a=x+1$$

Divide both sides by -y.

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

Dividing by -y undoes the multiplication by -y.

$$a=\frac{x+1}{-y}$$

Divide x+1 by -y.

$$a=-\frac{x+1}{y}$$

Solve for x

$$x=-ay-1$$

$$a=0\text{ or }y\neq -\frac{1}{a}$$