What are Separable Differential Equations? Definition & Explanation

📚 What are Separable Differential Equations?

A separable differential equation is one that can be written in the form:

$\frac{dy}{dx} = f(x)g(y)$

This means you can separate the variables $x$ and $y$ onto different sides of the equation. The general strategy is to get all terms involving $y$ on one side with $dy$, and all terms involving $x$ on the other side with $dx$. Then, you integrate both sides.

🧮 Steps to Solve Separable Differential Equations

• ➗ Separate the Variables: Algebraically manipulate the equation to get all $y$ terms with $dy$ on one side and all $x$ terms with $dx$ on the other side. This should result in an equation of the form: $h(y) dy = f(x) dx$.
• ➕ Integrate Both Sides: Integrate both sides of the separated equation: $\int h(y) dy = \int f(x) dx$. This gives you $H(y) = F(x) + C$, where $H(y)$ and $F(x)$ are the antiderivatives of $h(y)$ and $f(x)$, respectively, and $C$ is the constant of integration.
• 🔑 Solve for $y$: If possible, solve the resulting equation for $y$ to obtain the explicit solution $y = G(x)$. If solving for $y$ is difficult or impossible, the solution can be left in implicit form, $H(y) = F(x) + C$.

🧪 Example: Solving a Separable Differential Equation

Let's solve the differential equation:

$\frac{dy}{dx} = x y$

1. Separate the variables:$\frac{dy}{y} = x dx$
2. Integrate both sides:$\int \frac{dy}{y} = \int x dx$
   $\ln|y| = \frac{1}{2}x^2 + C$
3. Solve for $y$:$|y| = e^{\frac{1}{2}x^2 + C} = e^{\frac{1}{2}x^2}e^C$
   $y = \pm e^C e^{\frac{1}{2}x^2}$
   Since $\pm e^C$ is just another constant, we can write:
   $y = K e^{\frac{1}{2}x^2}$, where $K$ is an arbitrary constant.

📝 Practice Quiz

1. Solve $\frac{dy}{dx} = 2x$.
2. Solve $\frac{dy}{dx} = y$.
3. Solve $\frac{dy}{dx} = x^2y$.
4. Solve $\frac{dy}{dx} = \frac{x}{y}$.
5. Solve $\frac{dy}{dx} = e^x y$.
6. Solve $\frac{dy}{dx} = \frac{y}{x}$.
7. Solve $\frac{dy}{dx} = x\sqrt{y}$.