Solve y^{prime}=(x^2-x)d/dx^2-x^2-2d/dx(x^2-x)

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

Solve for d (complex solution)

$$d=-\frac{2x\left(1-2x\right)}{4x^{2}-3x+1}$$

$$x\neq 1\text{ and }x\neq \frac{3+\sqrt{7}i}{8}\text{ and }x\neq \frac{-\sqrt{7}i+3}{8}\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Variable d cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by \left(d-1\right)x^{2}.

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

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

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

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

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

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

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

Add 2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)\left(d-1\right)x^{2} to both sides.

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

Use the distributive property to multiply 2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x) by d-1.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+\left(2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)d-4x+2\right)x^{2}=x^{2}d-xd$$

Use the distributive property to multiply 2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)d-4x+2 by x^{2}.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}=x^{2}d-xd$$

Subtract x^{2}d from both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}-x^{2}d=-xd$$

Add xd to both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}-x^{2}d+xd=0$$

Add x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y) to both sides. Anything plus zero gives itself.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}-x^{2}d+xd=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Add 4x^{3} to both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}+2x^{2}-x^{2}d+xd=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+4x^{3}$$

Subtract 2x^{2} from both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-x^{2}d+xd=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+4x^{3}-2x^{2}$$

Combine all terms containing d.

$$\left(x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)x^{2}-x^{2}+x\right)d=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+4x^{3}-2x^{2}$$

The equation is in standard form.

$$\left(4x^{3}-3x^{2}+x\right)d=4x^{3}-2x^{2}$$

Divide both sides by 4x^{3}-3x^{2}+x.

$$\frac{\left(4x^{3}-3x^{2}+x\right)d}{4x^{3}-3x^{2}+x}=\frac{2\left(2x-1\right)x^{2}}{4x^{3}-3x^{2}+x}$$

Dividing by 4x^{3}-3x^{2}+x undoes the multiplication by 4x^{3}-3x^{2}+x.

$$d=\frac{2\left(2x-1\right)x^{2}}{4x^{3}-3x^{2}+x}$$

Divide 2\left(-1+2x\right)x^{2} by 4x^{3}-3x^{2}+x.

$$d=\frac{2x\left(2x-1\right)}{4x^{2}-3x+1}$$

Variable d cannot be equal to 1.

$$d=\frac{2x\left(2x-1\right)}{4x^{2}-3x+1}\text{, }d\neq 1$$

Solve for d

$$d=-\frac{2x\left(1-2x\right)}{4x^{2}-3x+1}$$

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

Steps for Solving Linear Equation

Variable d cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by \left(d-1\right)x^{2}.

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

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

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

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

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

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

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

Add 2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)\left(d-1\right)x^{2} to both sides.

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

Use the distributive property to multiply 2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x) by d-1.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+\left(2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)d-4x+2\right)x^{2}=x^{2}d-xd$$

Use the distributive property to multiply 2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)d-4x+2 by x^{2}.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}=x^{2}d-xd$$

Subtract x^{2}d from both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}-x^{2}d=-xd$$

Add xd to both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)-x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}-x^{2}d+xd=0$$

Add x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y) to both sides. Anything plus zero gives itself.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-4x^{3}+2x^{2}-x^{2}d+xd=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)$$

Add 4x^{3} to both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}+2x^{2}-x^{2}d+xd=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+4x^{3}$$

Subtract 2x^{2} from both sides.

$$dx^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)dx^{2}-x^{2}d+xd=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+4x^{3}-2x^{2}$$

Combine all terms containing d.

$$\left(x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+2\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}-x)x^{2}-x^{2}+x\right)d=x^{2}\frac{\mathrm{d}}{\mathrm{d}x}(y)+4x^{3}-2x^{2}$$

The equation is in standard form.

$$\left(4x^{3}-3x^{2}+x\right)d=4x^{3}-2x^{2}$$

Divide both sides by 4x^{3}-3x^{2}+x.

$$\frac{\left(4x^{3}-3x^{2}+x\right)d}{4x^{3}-3x^{2}+x}=\frac{2\left(2x-1\right)x^{2}}{4x^{3}-3x^{2}+x}$$

Dividing by 4x^{3}-3x^{2}+x undoes the multiplication by 4x^{3}-3x^{2}+x.

$$d=\frac{2\left(2x-1\right)x^{2}}{4x^{3}-3x^{2}+x}$$

Divide 2\left(-1+2x\right)x^{2} by 4x^{3}-3x^{2}+x.

$$d=\frac{2x\left(2x-1\right)}{4x^{2}-3x+1}$$

Variable d cannot be equal to 1.

$$d=\frac{2x\left(2x-1\right)}{4x^{2}-3x+1}\text{, }d\neq 1$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{4+4d-7d^{2}}+3d-2}{8\left(d-1\right)}\text{, }&d\neq 1\\x=\frac{\sqrt{4+4d-7d^{2}}+3d-2}{8\left(d-1\right)}\text{, }&d\neq 1\text{ and }d\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{4+4d-7d^{2}}+3d-2}{8\left(d-1\right)}\text{, }&d\neq 1\text{ and }d\geq \frac{2-4\sqrt{2}}{7}\text{ and }d\leq \frac{4\sqrt{2}+2}{7}\\x=\frac{\sqrt{4+4d-7d^{2}}+3d-2}{8\left(d-1\right)}\text{, }&d\neq 0\text{ and }d\neq 1\text{ and }d\geq \frac{2-4\sqrt{2}}{7}\text{ and }d\leq \frac{4\sqrt{2}+2}{7}\end{matrix}\right.$$