Solve 3x^2dx-2ydy=0

$$3 x ^ { 2 } d x - 2 y d y = 0$$

Solve for d (complex solution)

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

Steps for Solving Linear Equation

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

$$3x^{3}d-2ydy=0$$

Multiply y and y to get y^{2}.

$$3x^{3}d-2y^{2}d=0$$

Combine all terms containing d.

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

Divide 0 by 3x^{3}-2y^{2}.

$$d=0$$

Solve for d

$$\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&x=\frac{\sqrt[3]{2}\times \left(3y\right)^{\frac{2}{3}}}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$3x^{3}d-2ydy=0$$

Multiply y and y to get y^{2}.

$$3x^{3}d-2y^{2}d=0$$

Combine all terms containing d.

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

Divide 0 by 3x^{3}-2y^{2}.

$$d=0$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{\sqrt[3]{2}e^{\frac{2\pi i}{3}}\times \left(3y\right)^{\frac{2}{3}}}{3}\text{; }x=\frac{\sqrt[3]{2}\times \left(3y\right)^{\frac{2}{3}}}{3}\text{; }x=\frac{\sqrt[3]{2}e^{\frac{4\pi i}{3}}\times \left(3y\right)^{\frac{2}{3}}}{3}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&d=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{\sqrt[3]{2}\times \left(3y\right)^{\frac{2}{3}}}{3}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&d=0\end{matrix}\right.$$