Separable Differential Equations- Solving Methods
What Are Separable Differential Equations?
A separable differential equation is one where you can rearrange the terms so that all the x stuff stays on one side and all the y stuff stays on the other. That's it. That's the whole idea.
These equations show up everywhere in physics, engineering, and biology. Population growth, radioactive decay, cooling coffee, falling objects—all of these are modeled with separable equations. If you can't solve these, you're going to hit a wall fast in any applied math course.
The Form You Need to Memorize
Every separable differential equation can be written as:
dy/dx = g(x) · h(y)
Notice the structure. The derivative is the product of two functions—one depends only on x, the other only on y. If your equation doesn't look like this, it's not separable. Move on.
The Solving Method (Step by Step)
Here's how you actually solve these things. No magic, just algebra done in the right order.
Step 1: Separate the Variables
Start with dy/dx = g(x) · h(y). Rearrange it to:
dy / h(y) = g(x) dx
Get all y terms with dy on one side. Get all x terms with dx on the other. If you can't do this cleanly, you have the wrong type of equation.
Step 2: Integrate Both Sides
Now you have two integrals:
∫[dy / h(y)] = ∫[g(x) dx]
Integrate each side separately. Add the constant of integration—you only need one, usually written as C on the right side.
Step 3: Solve for y (If Possible)
After integration, you get an implicit solution. Sometimes you can solve for y explicitly. Sometimes you can't. That's fine—implicit solutions count.
Working Example
Let's do one completely so you see how this flows.
Solve: dy/dx = xy
Step 1: Separate the variables
dy/dx = x · y
dy / y = x dx
Step 2: Integrate both sides
∫(1/y) dy = ∫x dx
ln|y| = x²/2 + C
Step 3: Solve for y
eln|y| = e^(x²/2 + C)
|y| = e^(x²/2) · e^C
y = Ce^(x²/2)
That's your general solution. C is any constant. Notice how the absolute value disappeared when we absorbed it into C—that's a standard move.
Initial Value Problems
Most textbook problems give you an initial condition like y(0) = 1. Use it to find the specific value of C.
Take the general solution, plug in x=0 and y=1, solve for C, then write the particular solution. That's it. No shortcuts here—you just substitute and solve.
Common Mistakes That Cost You Points
- Trying to separate an equation that isn't separable. Check the form first.
- Forgetting the constant of integration. Always include C after integrating.
- Losing a sign during separation. Be careful when moving terms across the equals sign.
- Not checking your answer by differentiating. Do this. Catch your own errors before your professor does.
Separable vs. Other Differential Equations
You need to know when separable is the right tool. Here's how it compares:
| Equation Type | Can You Separate? | Solving Method |
|---|---|---|
| Separable (dy/dx = g(x)·h(y)) | Yes, that's the definition | Integration after separation |
| Linear (dy/dx + P(x)y = Q(x)) | No | Integrating factor |
| Exact equations | No | Check Mdx + Ndy = 0 for exactness |
| Homogeneous | Sometimes after substitution | Let v = y/x, then separate |
| Bernoulli (dy/dx + P(x)y = Q(x)yⁿ) | No | Substitute u = y^(1-n) |
Practice Problems to Try
You learn by doing, not reading. Work through these in order:
- dy/dx = y² · cos(x) — separate, integrate, solve for y
- dy/dx = (3x² + 4) / y² — same process
- dy/dx = x · sqrt(y) — pay attention to the square root when integrating
- dy/dx = e^(x+y) — use the exponential property: e^(x+y) = e^x · e^y
- An IVP: dy/dx = 2x, y(0) = 3 — find the particular solution
The Bottom Line
Separable differential equations are the easiest type to solve once you understand the process. Separate, integrate, solve for y. Three steps. Memorize the form, practice the algebra, and you'll get these right every time.
If you're struggling, it's probably because you're trying to skip steps or guessing instead of following the method. Slow down. Write out each step. The answer will come.