First Order Differential Equations- Complete Tutorial

First Order Differential Equations: What You Actually Need to Know

A first order differential equation is any equation that contains a first derivative (dy/dx) and nothing higher. That's the whole definition. Nothing fancy. If you see dy/dx, you're dealing with first order.

These equations show up everywhere in physics, engineering, and anywhere rates of change matter. Population growth, cooling objects, electrical circuits, chemical reactions—all of them use first order differential equations. You need to know how to solve them.

The General Form

Every first order differential equation can be written as:

M(x, y)dx + N(x, y)dy = 0

Or equivalently:

dy/dx = f(x, y)

That's it. Everything else is just figuring out which specific type you're working with and applying the right technique.

The Four Types You Must Know

There are four main types of first order equations. Learn to recognize each one.

1. Separable Equations

These are the easiest. A separable equation is one where x and y can be separated onto opposite sides of the equation.

General form: dy/dx = g(x)h(y)

The solution method: move everything with y to one side, everything with x to the other, then integrate both sides.

2. Linear Equations

Linear first order equations have this specific form:

dy/dx + P(x)y = Q(x)

The y term and its derivative are only to the first power—no y², no sin(y), nothing weird. Just y multiplied by functions of x.

You solve these using an integrating factor.

3. Exact Equations

An equation M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x.

If that condition holds, there's a potential function ψ(x,y) where ∂ψ/∂x = M and ∂ψ/∂y = N. Set ψ(x,y) = C and you're done.

4. Bernoulli Equations

These look like linear equations but have a yⁿ term:

dy/dx + P(x)y = Q(x)yⁿ

They look harder than they are. Use a substitution to convert them into linear equations.

How to Solve Each Type

Separable Equations: Step by Step

Here's exactly what you do:

1. Write dy/dx = f(x)g(y)
2. Separate: dy/g(y) = f(x)dx
3. Integrate both sides
4. Solve for y if possible

Example: dy/dx = xy

Separate: dy/y = x dx

Integrate: ln|y| = x²/2 + C

Solve: y = Ce^(x²/²)

Linear Equations: The Integrating Factor Method

For dy/dx + P(x)y = Q(x):

1. Find the integrating factor: μ(x) = e^(∫P(x)dx)
2. Multiply the entire equation by μ(x)
3. The left side becomes d/dx[μ(x)y]
4. Integrate both sides
5. Solve for y

Example: dy/dx + 2y = 4x

Integrating factor: μ = e^(∫2dx) = e^(2x)

Multiply: e^(2x)dy/dx + 2e^(2x)y = 4xe^(2x)

Left side is d/dx[e^(2x)y]

Integrate: e^(2x)y = ∫4xe^(2x)dx = 2xe^(2x) - e^(2x) + C

Solve: y = 2x - 1 + Ce^(-2x)

Exact Equations: Finding the Potential Function

Check if ∂M/∂y = ∂N/∂x first. If true:

1. Integrate M with respect to x: ψ = ∫Mdx + h(y)
2. Find h'(y) by differentiating ψ and setting it equal to N
3. Integrate h'(y) to get h(y)
4. Set ψ(x,y) = C

Bernoulli Equations: The Substitution

For dy/dx + P(x)y = Q(x)yⁿ:

1. Let u = y^(1-n)
2. Then du/dx = (1-n)y^(-n)dy/dx
3. Substitute into the original equation
4. You get a linear equation in u
5. Solve using the integrating factor method
6. Substitute back to get y

Comparison of Solution Methods

Initial Value Problems (IVPs)

An IVP is just a differential equation plus an initial condition like y(x₀) = y₀.

You solve this in two steps:

1. Solve the differential equation to get the general solution
2. Use the initial condition to find the constant C

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

Integrate: y = x² + C

Use y(0) = 3: 3 = 0 + C, so C = 3

Solution: y = x² + 3

Getting Started: Your Action Plan

When you see a first order differential equation, work through this checklist:

• Can you separate variables? If yes, do that first. It's the fastest method.
• Is it linear? Check for y term with no products or powers. Use integrating factor.
• Is it exact? Test ∂M/∂y = ∂N/∂x. If true, find the potential function.
• Is it Bernoulli? Look for y raised to a power on one side. Use substitution.
• Nothing matches? Try a substitution to transform it into one of these forms.

Common Mistakes That Cost You Points

• Forgetting the constant of integration. Always include +C after integration. Every time.
• Not checking if the equation is exact. Jumping to other methods when exact would work faster.
• Messy algebra. The integrating factor method fails if you make algebra errors. Write every step.
• Wrong substitution in Bernoulli. Remember: u = y^(1-n), not y^(n-1).
• Solving for y when you don't need to. Sometimes ψ(x,y) = C is a fine final answer.

Verifying Your Solution

Always check your work. Take your solution y(x) and calculate dy/dx. Plug both into the original equation. Both sides must match.

If they don't, you made an error somewhere. Go back and check your integration and algebra.

What Comes Next

Once you master first order equations, you're ready for second order linear differential equations. Those require a different toolkit—characteristic equations, undetermined coefficients, variation of parameters.

But first, lock down these four types. Practice until you can identify and solve each one without hesitation. That's the baseline for anything more advanced.