Solve partial/x-partial_{0}-sin(x)/x=-1

$$\frac { \partial } { x - \partial _ { 0 } } - \frac { \sin ( x ) } { x } = - 1$$

Solve for ∂

$$∂=-\frac{∂_{0}\sin(x)}{x}+\sin(x)+∂_{0}-x$$

$$x\neq ∂_{0}\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(-x+∂_{0}\right), the least common multiple of x-∂_{0},x.

$$-x∂-\left(-x+∂_{0}\right)\sin(x)=-x\left(-x+∂_{0}\right)$$

Use the distributive property to multiply -x+∂_{0} by \sin(x).

$$-x∂-\left(-x\sin(x)+∂_{0}\sin(x)\right)=-x\left(-x+∂_{0}\right)$$

To find the opposite of -x\sin(x)+∂_{0}\sin(x), find the opposite of each term.

$$-x∂+x\sin(x)-∂_{0}\sin(x)=-x\left(-x+∂_{0}\right)$$

Use the distributive property to multiply -x by -x+∂_{0}.

$$-x∂+x\sin(x)-∂_{0}\sin(x)=x^{2}-x∂_{0}$$

Subtract x\sin(x) from both sides.

$$-x∂-∂_{0}\sin(x)=x^{2}-x∂_{0}-x\sin(x)$$

Add ∂_{0}\sin(x) to both sides.

$$-x∂=x^{2}-x∂_{0}-x\sin(x)+∂_{0}\sin(x)$$

The equation is in standard form.

$$\left(-x\right)∂=∂_{0}\sin(x)-x\sin(x)-x∂_{0}+x^{2}$$

Divide both sides by -x.

$$\frac{\left(-x\right)∂}{-x}=\frac{x\left(-\sin(x)+x-∂_{0}\right)+∂_{0}\sin(x)}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$∂=\frac{x\left(-\sin(x)+x-∂_{0}\right)+∂_{0}\sin(x)}{-x}$$

Divide ∂_{0}\sin(x)+x\left(x-∂_{0}-\sin(x)\right) by -x.

$$∂=\frac{\left(∂_{0}-x\right)\left(-\sin(x)+x\right)}{x}$$