Solve y^prime=-a/(x+dx)^2

$$y ^ { \prime } = \frac { - a } { ( x + d x ) ^ { 2 } }$$

Solve for a

$$a=0$$

$$d\neq -1\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}\left(d+1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(d+1\right)^{2}.

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

Use the distributive property to multiply x^{2} by d^{2}+2d+1.

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

Use the distributive property to multiply x^{2}d^{2}+2x^{2}d+x^{2} by \frac{\mathrm{d}}{\mathrm{d}x}(y).

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

Swap sides so that all variable terms are on the left hand side.

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

The equation is in standard form.

$$-a=0$$

Divide 0 by -1.

$$a=0$$

Solve for d

$$d\neq -1$$

$$a=0\text{ and }x\neq 0$$