Linear First Order Differential Equation- Solutions

What Is a Linear First Order Differential Equation?

A linear first order differential equation has this form:

y' + P(x)y = Q(x)

That's it. No products of y and its derivatives. No functions multiplied together. Just a clean, linear relationship between y and its derivative.

The key feature is that y and y' appear to the first power only. If you see y² or sin(y), you're dealing with something else entirely.

The Integrating Factor Method

This is the standard technique. It works every time for this equation type. Here's how it goes.

Step 1: Get It Into Standard Form

Your equation must look like y' + P(x)y = Q(x).

If you have 2xy + y' = x², rearrange it:

y' + 2xy = x²

Now P(x) = 2x and Q(x) = x².

Step 2: Find the Integrating Factor

Calculate μ(x) = e^∫P(x)dx

Using our example where P(x) = 2x:

μ(x) = e^{∫2x dx} = e^x²

Step 3: Multiply Everything

Multiply the entire standard form equation by μ(x):

e^x²y' + 2xe^x²y = x²e^x²

The left side is not random. It always simplifies to d/dx[μ(x)·y].

Step 4: Integrate Both Sides

d/dx[μ(x)·y] = Q(x)·μ(x)

Integrate:

μ(x)·y = ∫Q(x)·μ(x)dx + C

Then solve for y:

y = [∫Q(x)·μ(x)dx + C] / μ(x)

Complete Worked Example

Solve: y' + 3y = 6

Step 1: Already in standard form. P(x) = 3, Q(x) = 6.

Step 2: Integrating factor:

μ = e^∫3dx = e^3x

Step 3: Multiply through:

e^3xy' + 3e^3xy = 6e^3x

Step 4: Recognize the derivative and integrate:

d/dx[e^3xy] = 6e^3x

e^3xy = ∫6e^3xdx

e^3xy = 6 · (1/3)e^3x + C

e^3xy = 2e^3x + C

Step 5: Solve for y:

y = 2 + Ce^-3x

That's your general solution. The constant C gets determined by initial conditions if you have them.

Quick Reference Table

Common Mistakes That Will Cost You Points

• Forgetting to multiply both sides by the integrating factor. You must multiply the entire equation, not just the left side.
• Solving the integral wrong in step 2. Check your antiderivative before moving on.
• Dropping the constant after integration. That C is there for a reason.
• Not simplifying the final answer. y = e^x + Ce^x simplifies to y = (1+C)e^x.

How to Get Started With Any Problem

When you see a differential equation, run this checklist:

1. Is it linear? Check if y and y' are to the first power with no products between them.
2. Rearrange to standard form if needed. Get y' alone on one side.
3. Identify P(x) and Q(x) from the standard form.
4. Compute μ = e^∫P(x)dx. Don't forget the +C is not included here.
5. Multiply the whole equation by μ.
6. Integrate both sides. The left side becomes μy.
7. Solve for y and simplify.

Practice this sequence until it becomes automatic. Most errors come from skipping steps or rushing through the integrating factor calculation.

Initial Value Problems

If you have an initial condition like y(0) = 1, plug it into your general solution to find C.

Using our example y = 2 + Ce^-3x with y(0) = 1:

1 = 2 + Ce⁰

1 = 2 + C

C = -1

The particular solution is y = 2 - e^-3x

That's all there is to it. The integrating factor method is mechanical once you understand why it works. Memorize the steps, verify each one, and always check your answer by taking its derivative and substituting back into the original equation.