⚠ 2x is read as the product 2*x: for a power, write 2^x‖3y is read as the product 3*y: for a power, write 3^y‖1/2*(dy)/(dx)=(2x)/(3y) is read as 1/2*y'=(2x)/(3y): use dif(expression, variable) for a derivative with respect to that variable, use (numerator)/(denominator) for a quotient

Solve the separable equation $\frac{1}{2}·\mathrm{y\text{'}}=\frac{2x}{3y}$, where $y$ is a function of $x$ and $y$ ≠ 0

⇥Let's find the domain of $\frac{4x}{3}$

The function of $x$ is defined for $x\in ℝ$

Multiply by $y$ and separate the variables: $y·\mathrm{dy}=\frac{4x}{3}·\mathrm{dx}$

⇥Let's compute the integral: $\int (\frac{4x}{3})dx$

⇥ $\frac{4}{3}·\frac{x^2}{2}$

⇥ $\frac{4x^2}{6}$

⇥ $\frac{2x^2}{3}$

Integrating both sides gives $\frac{y^2}{2}=\frac{2x^2}{3}+C$, where $C$ is an arbitrary real constant

Multiply by 2 and rename $2C$ as $C$: $y^2=\frac{4x^2}{3}+C$

The two real branches have a fixed sign on each interval where the original right side is real and $\frac{4x^2}{3}+C$ > 0; $y$ = 0 is excluded

▶Solution:   $y=\{-\sqrt{\frac{4x^2}{3}+C},\sqrt{\frac{4x^2}{3}+C}\}$