Solving Separable Differential Equations- Y' Only

What Is a Separable Differential Equation?

A separable differential equation is one where you can isolate all y terms on one side and all x terms on the other side. That's it. That's the whole trick.

If you can rearrange your equation into the form:

g(y) dy/dx = f(x)

Then congratulations — you have a separable equation. You can solve it by integrating both sides.

The General Solution Method

Here's the process:

1. Start with dy/dx = f(x, y)
2. Rewrite as dy/dx = g(x)/h(y) or h(y) dy/dx = g(x)
3. Multiply both sides to separate variables: h(y) dy = g(x) dx
4. Integrate both sides
5. Solve for y if you can

That's four steps. Memorize them.

Working Through an Example

Let's solve:

dy/dx = x/y

Step 1: Separate the variables. Multiply both sides by y and dx:

y dy = x dx

Step 2: Integrate both sides:

∫y dy = ∫x dx

½y² = ½x² + C

Step 3: Solve for y:

y² = x² + C₁

y = ±√(x² + C₁)

Done. That's the entire solution.

Another Example With Exponential Terms

Solve: dy/dx = 2y

This looks simple, but it trips people up.

Step 1: Separate — divide both sides by y:

(1/y) dy/dx = 2

Rewrite as: dy/y = 2 dx

Step 2: Integrate:

ln|y| = 2x + C

Step 3: Exponentiate:

|y| = e^(2x + C) = e^C · e^(2x)

y = Ce^(2x)

Notice the absolute value disappears when you consolidate the constant. This is standard — just use C instead of ±C.

Common Mistakes That Cost You Points

• Forgetting the constant of integration. Always include +C after integrating. Missing it means you've lost a family of solutions.
• Not separating correctly. If you can't get y terms alone on one side and x terms alone on the other, you don't have a separable equation. Check your algebra.
• Losing solutions during separation. When you divide by something, you might be discarding a solution where that something equals zero. For dy/dx = 2y, y = 0 is also a valid solution — and it gets lost if you divide by y.
• Integration errors. ln|y| integrates to y, but y² integrates to y³/3. Basic errors sink people here.

Separable vs. Non-Separable: How to Tell

Quick test: can you rearrange into f(y) dy = g(x) dx?

If yes → separable. If no → not separable (at least not in this form).

Examples:

dy/dx = x + y → NOT separable. You can't separate x and y when they're added together.

dy/dx = xy → Separable. Rewrite as dy/y = x dx.

dy/dx = (x + y)/x → NOT separable in current form. Simplifies to 1 + y/x, which still mixes variables.

Quick Reference Cheat Sheet

Practical How-To: Solving Any Separable Equation

Here's your step-by-step workflow:

1. Identify the differential equation type. Make sure you're actually working with a separable one before wasting time.

2. Rearrange to separate variables. Get all y terms on the left with dy, all x terms on the right with dx. Use algebra — multiply, divide, factor until the separation is clean.

3. Integrate both sides. Don't forget the constant. If the integrals look ugly, double-check your separation.

4. Solve for y explicitly if required. Sometimes C stays implicit and that's fine. Other problems explicitly ask for y = f(x). If you can solve algebraically, do it.

5. Check for lost solutions. When you divide during separation, note any values that make the divisor zero. These might be valid solutions.

Initial Value Problems (IVPs)

Often you'll get a condition like y(0) = 1. This lets you find the specific value of C.

Example:

dy/dx = 2x, y(0) = 3

Step 1: Separate: dy = 2x dx

Step 2: Integrate: y = x² + C

Step 3: Use the initial condition: 3 = 0² + C → C = 3

Answer: y = x² + 3

That's it. Plug in, solve for C, done.

What You Need to Memorize

• The separation process: get y terms with dy, x terms with dx
• How to integrate basic forms: ∫dy/y = ln|y| + C, ∫x^n dx = x^(n+1)/(n+1) + C
• That you must include +C
• That dividing by a function might discard a solution

If you know these four things, you can solve any separable differential equation you'll encounter.

The Bottom Line

Separable differential equations are the easiest type to solve. The method is straightforward: separate, integrate, solve. There's no trick, no special insight required. You either follow the steps or you don't.

Master this before moving on to linear equations or exact equations. Those actually require some thought.