Solve y=sqrt[3]{x}dx+2

$$y = \sqrt[ 3 ] { x } d x + 2$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=-x^{-\frac{4}{3}}\left(2-y\right)\text{, }&x\neq 0\\d\in \mathrm{C}\text{, }&y=2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\sqrt[3]{x}dx+2=y$$

Subtract 2 from both sides.

$$\sqrt[3]{x}dx=y-2$$

The equation is in standard form.

$$\sqrt[3]{x}xd=y-2$$

Divide both sides by \sqrt[3]{x}x.

$$\frac{\sqrt[3]{x}xd}{\sqrt[3]{x}x}=\frac{y-2}{\sqrt[3]{x}x}$$

Dividing by \sqrt[3]{x}x undoes the multiplication by \sqrt[3]{x}x.

$$d=\frac{y-2}{\sqrt[3]{x}x}$$

Divide y-2 by \sqrt[3]{x}x.

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

Solve for d

$$\left\{\begin{matrix}d=-\frac{2-y}{x^{\frac{4}{3}}}\text{, }&x\neq 0\\d\in \mathrm{R}\text{, }&y=2\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\sqrt[3]{x}dx+2=y$$

Subtract 2 from both sides.

$$\sqrt[3]{x}dx=y-2$$

The equation is in standard form.

$$\sqrt[3]{x}xd=y-2$$

Divide both sides by \sqrt[3]{x}x.

$$\frac{\sqrt[3]{x}xd}{\sqrt[3]{x}x}=\frac{y-2}{\sqrt[3]{x}x}$$

Dividing by \sqrt[3]{x}x undoes the multiplication by \sqrt[3]{x}x.

$$d=\frac{y-2}{\sqrt[3]{x}x}$$

Divide y-2 by \sqrt[3]{x}x.

$$d=\frac{y-2}{x^{\frac{4}{3}}}$$

Solve for x (complex solution)

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

Solve for x

$$\left\{\begin{matrix}x=\sqrt[4]{-\left(\frac{2-y}{d}\right)^{3}}\text{; }x=-\sqrt[4]{-\left(\frac{2-y}{d}\right)^{3}}\text{, }&\left(y\leq 2\text{ and }d<0\right)\text{ or }\left(y\geq 2\text{ and }d>0\right)\\x\in \mathrm{R}\text{, }&y=2\text{ and }d=0\end{matrix}\right.$$