Solve e^{x}(y^3+xy^3+1)dx+3y^2(xe^x-6)dy=0

$$e ^ { x } ( y ^ { 3 } + x y ^ { 3 } + 1 ) d x + 3 y ^ { 2 } ( x e ^ { x } - 6 ) d y = 0$$

Solve for d (complex solution)

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(y=-\sqrt[3]{x}e^{\frac{x+2\pi i}{3}}\left(x^{2}e^{x}+4xe^{x}-18\right)^{-\frac{1}{3}}\text{ or }y=i\sqrt[3]{x}e^{\frac{x}{3}+\frac{\pi i}{2}}\left(x^{2}e^{x}+4xe^{x}-18\right)^{-\frac{1}{3}}\text{ or }y=\sqrt[3]{x}e^{\frac{x+\pi i}{3}}\left(x^{2}e^{x}+4xe^{x}-18\right)^{-\frac{1}{3}}\right)\text{ and }x^{2}e^{x}+4xe^{x}-18\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$e^{x}\left(y^{3}+xy^{3}+1\right)dx+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply e^{x} by y^{3}+xy^{3}+1.

$$\left(e^{x}y^{3}+e^{x}xy^{3}+e^{x}\right)dx+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply e^{x}y^{3}+e^{x}xy^{3}+e^{x} by d.

$$\left(e^{x}y^{3}d+e^{x}xy^{3}d+e^{x}d\right)x+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply e^{x}y^{3}d+e^{x}xy^{3}d+e^{x}d by x.

$$e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply 3y^{3} by xe^{x}-6.

$$e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx+\left(3y^{3}xe^{x}-18y^{3}\right)d=0$$

Use the distributive property to multiply 3y^{3}xe^{x}-18y^{3} by d.

$$e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx+3y^{3}xe^{x}d-18y^{3}d=0$$

Combine e^{x}y^{3}dx and 3y^{3}xe^{x}d to get 4e^{x}y^{3}dx.

$$4e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx-18y^{3}d=0$$

Combine all terms containing d.

$$\left(4e^{x}y^{3}x+e^{x}y^{3}x^{2}+e^{x}x-18y^{3}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{3}e^{x}+4xy^{3}e^{x}+xe^{x}-18y^{3}\right)d=0$$

Divide 0 by 4e^{x}y^{3}x+e^{x}y^{3}x^{2}+e^{x}x-18y^{3}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&y=-\frac{\sqrt[3]{x}e^{\frac{x}{3}}}{\sqrt[3]{x^{2}e^{x}+4xe^{x}-18}}\text{ and }x^{2}e^{x}+4xe^{x}-18\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$e^{x}\left(y^{3}+xy^{3}+1\right)dx+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply e^{x} by y^{3}+xy^{3}+1.

$$\left(e^{x}y^{3}+e^{x}xy^{3}+e^{x}\right)dx+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply e^{x}y^{3}+e^{x}xy^{3}+e^{x} by d.

$$\left(e^{x}y^{3}d+e^{x}xy^{3}d+e^{x}d\right)x+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply e^{x}y^{3}d+e^{x}xy^{3}d+e^{x}d by x.

$$e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx+3y^{3}\left(xe^{x}-6\right)d=0$$

Use the distributive property to multiply 3y^{3} by xe^{x}-6.

$$e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx+\left(3y^{3}xe^{x}-18y^{3}\right)d=0$$

Use the distributive property to multiply 3y^{3}xe^{x}-18y^{3} by d.

$$e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx+3y^{3}xe^{x}d-18y^{3}d=0$$

Combine e^{x}y^{3}dx and 3y^{3}xe^{x}d to get 4e^{x}y^{3}dx.

$$4e^{x}y^{3}dx+e^{x}y^{3}dx^{2}+e^{x}dx-18y^{3}d=0$$

Combine all terms containing d.

$$\left(4e^{x}y^{3}x+e^{x}y^{3}x^{2}+e^{x}x-18y^{3}\right)d=0$$

The equation is in standard form.

$$\left(x^{2}y^{3}e^{x}+4xy^{3}e^{x}+xe^{x}-18y^{3}\right)d=0$$

Divide 0 by 4e^{x}y^{3}x+e^{x}y^{3}x^{2}+e^{x}x-18y^{3}.

$$d=0$$