Solve Xd^2y/dx^2+2partialdy/dx=12x^2

$$X \frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { \partial d y } { d x } = 12 x ^ { 2 }$$

Solve for X

$$X\in \mathrm{R}$$

$$y=\frac{6x^{3}}{∂}\text{ and }∂\neq 0\text{ and }x\neq 0\text{ and }d\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by dx.

$$X\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}dx+2∂dy=12x^{2}dx$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$X\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}dx+2∂dy=12x^{3}d$$

Subtract 2∂dy from both sides.

$$X\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}dx=12x^{3}d-2∂dy$$

The equation is in standard form.

$$0=12dx^{3}-2dy∂$$

This is false for any X.

$$X\in$$

Solve for d

$$d\neq 0$$

$$y=\frac{6x^{3}}{∂}\text{ and }∂\neq 0\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by dx.

$$X\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}dx+2∂dy=12x^{2}dx$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$X\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}dx+2∂dy=12x^{3}d$$

Subtract 12x^{3}d from both sides.

$$X\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}dx+2∂dy-12x^{3}d=0$$

Combine all terms containing d.

$$\left(X\frac{\mathrm{d}(y)}{\mathrm{d}x^{2}}x+2∂y-12x^{3}\right)d=0$$

The equation is in standard form.

$$\left(2y∂-12x^{3}\right)d=0$$

Divide 0 by 2∂y-12x^{3}.

$$d=0$$

Variable d cannot be equal to 0.

$$d\in \emptyset$$