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

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

Solve for a

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

$$y\neq 0\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by xy, the least common multiple of y,x.

$$xy\frac{\mathrm{d}}{\mathrm{d}x}(y)-axx=y\left(x+1\right)$$

Multiply x and x to get x^{2}.

$$xy\frac{\mathrm{d}}{\mathrm{d}x}(y)-ax^{2}=y\left(x+1\right)$$

Use the distributive property to multiply y by x+1.

$$xy\frac{\mathrm{d}}{\mathrm{d}x}(y)-ax^{2}=yx+y$$

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

$$-ax^{2}=yx+y-xy\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

The equation is in standard form.

$$\left(-x^{2}\right)a=xy+y$$

Divide both sides by -x^{2}.

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

Dividing by -x^{2} undoes the multiplication by -x^{2}.

$$a=\frac{xy+y}{-x^{2}}$$

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

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{y\left(y-4a\right)}+y}{2a}\text{; }x=-\frac{-\sqrt{y\left(y-4a\right)}+y}{2a}\text{, }&a\neq 0\text{ and }y\neq 0\\x=-1\text{, }&a=0\text{ and }y\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{y\left(y-4a\right)}+y}{2a}\text{; }x=-\frac{-\sqrt{y\left(y-4a\right)}+y}{2a}\text{, }&\left(y<0\text{ or }y\geq 4a\right)\text{ and }y\neq 0\text{ and }a\neq 0\text{ and }\left(y>0\text{ or }y\leq 4a\right)\\x=-1\text{, }&a=0\text{ and }y\neq 0\end{matrix}\right.$$