DE-BOOK Separation of Variables

Chapter 5 Separation of Variables

📝: 🎧 Listen.

When a differential equation is both first-order and separable, we can reliably find its general solution using the separation of variables method. At its core, this technique involves rewriting the equation so that all \(y\) terms are grouped on one side and all \(x\) terms on the other. Once the two sides are separated, integrating them yields the general solution.

🔗

Once you verify that the separation of variables method applies, the strategy can be summarized by a 3-step workflow:

🔗

Separate the \(x\) & \(y\) variables ➜ Integrate the separated equation ➜ Isolate \(y\text{.}\)

🔗

This chapter guides you through this process, starting with what it means for an equation to be separable and finishing with how to find the general solution.

🔗

🗝️ Key Takeaways...

A first-order differential equation is Separable when the right-hand side is a product of an \(x\)-only part and a \(y\)-only part:
\begin{equation*} \dfrac{dy}{dx}=f(x)\cdot g(y)\text{.} \end{equation*}

🔗
The separation of variables (SOV) method applies only to \(\textit{first-order, separable}\) differential equations.

🔗
Method Workflow:
1. verify separability and separate variables,
  
  🔗
2. integrate,
  
  🔗
3. solve for \(y\) (if possible) and combine constants.
  
  🔗
🔗
🔗
It can be helpful to apply initial conditions to find the constant \(c\) before any step that introduces \(\pm\text{.}\)
🔗

🔗
Absolute values often arise from logarithms; resolve \(|\cdot|\) by introducing \(\pm\) and then absorbing the sign into the constant when appropriate.
🔗

🔗

🔗

Prev Top Next