First Order Differential Equation

First Order Differential Equation

An ordinary differential equation of order 1. That is, a differential equation in which the highest derivative is a first derivative. For example, y' + xy = 1 is a first order differential equation.

Key Formula

\frac{dy}{dx} = f(x,\, y)

Where:

$y$ = The unknown function of x that you are solving for
$x$ = The independent variable
$\frac{dy}{dx}$ = The first derivative of y with respect to x
$f(x, y)$ = A given function that defines the relationship between x, y, and dy/dx

Worked Example

Problem: Solve the first order differential equation dy/dx = 3x² with the initial condition y(0) = 5.

Step 1: Identify the type. The right-hand side depends only on x, so this is a directly integrable equation. Rewrite it as:

dy = 3x^2\, dx

Step 2: Integrate both sides:

\int dy = \int 3x^2\, dx \implies y = x^3 + C

Step 3: Apply the initial condition y(0) = 5 to find C:

5 = (0)^3 + C \implies C = 5

Step 4: Write the particular solution:

y = x^3 + 5

Answer: The solution is y = x³ + 5.

Another Example

This example uses the integrating factor method for a first order linear equation, unlike the first example which was solved by direct integration. It shows the most common technique students need for first order equations.

Problem: Solve the first order linear differential equation dy/dx + 2y = 6 with the initial condition y(0) = 1.

Step 1: Identify the form. This is a first order linear equation dy/dx + P(x)y = Q(x) with P(x) = 2 and Q(x) = 6. Find the integrating factor:

\mu(x) = e^{\int 2\, dx} = e^{2x}

Step 2: Multiply both sides of the equation by the integrating factor:

e^{2x}\frac{dy}{dx} + 2e^{2x}y = 6e^{2x}

Step 3: Recognize the left side as the derivative of the product μ(x)·y, then integrate both sides:

\frac{d}{dx}\bigl(e^{2x}y\bigr) = 6e^{2x} \implies e^{2x}y = \int 6e^{2x}\, dx = 3e^{2x} + C

Step 4: Divide both sides by the integrating factor:

y = 3 + Ce^{-2x}

Step 5: Apply the initial condition y(0) = 1:

1 = 3 + C \implies C = -2 \implies y = 3 - 2e^{-2x}

Answer: The solution is y = 3 − 2e^(−2x).

Frequently Asked Questions

What is the difference between a first order and a second order differential equation?

A first order differential equation contains derivatives up to dy/dx (the first derivative) and no higher. A second order differential equation contains a second derivative d²y/dx² as the highest derivative. The order tells you how many times you need to integrate (broadly speaking) and how many initial/boundary conditions are needed for a unique solution — one for first order, two for second order.

How do you solve a first order differential equation?

The method depends on the type. If variables can be separated, use separation of variables. For linear first order equations of the form dy/dx + P(x)y = Q(x), use the integrating factor method. Other types include exact equations and homogeneous equations, each with their own technique. Always start by identifying which form the equation fits.

What are some real-life examples of first order differential equations?

First order differential equations model many real phenomena. Radioactive decay and population growth follow dy/dt = ky. Newton's law of cooling, where an object's temperature changes proportionally to the difference from the environment, is a first order equation. Mixing problems involving concentration in tanks and simple electrical circuits (RC circuits) are also modeled by first order equations.

First Order Differential Equation vs. Second Order Differential Equation

	First Order Differential Equation	Second Order Differential Equation
Highest derivative	First derivative (dy/dx)	Second derivative (d²y/dx²)
General form	F(x, y, y') = 0	F(x, y, y', y'') = 0
Initial conditions needed	One (e.g., y(0) = a)	Two (e.g., y(0) = a and y'(0) = b)
Common solution methods	Separation of variables, integrating factor, exact equations	Characteristic equation, undetermined coefficients, variation of parameters
Typical applications	Exponential growth/decay, RC circuits, cooling	Vibrations, springs, RLC circuits, beam deflection

Why It Matters

First order differential equations are the starting point for all differential equations courses and appear throughout science and engineering. You encounter them when modeling exponential growth, radioactive decay, Newton's law of cooling, and basic circuit analysis. Mastering the techniques for first order equations — especially separation of variables and integrating factors — builds the foundation for tackling the higher order and partial differential equations you will meet later.

Common Mistakes

Mistake: Forgetting to include the constant of integration C when finding the general solution.

Correction: Every indefinite integration produces an arbitrary constant. Without C, you have only one specific solution rather than the full family of solutions. The constant is then determined by an initial condition.

Mistake: Incorrectly computing the integrating factor by forgetting to exponentiate. Students write μ = ∫P(x) dx instead of μ = e^(∫P(x) dx).

Correction: The integrating factor is always an exponential: μ(x) = e^(∫P(x) dx). Writing just the integral without the exponential will produce the wrong multiplier and an incorrect solution.

Related Terms

Ordinary Differential Equation — The broader category that includes first order equations
Order of a Differential Equation — Defines the order based on the highest derivative
Differential Equation — The general concept of equations involving derivatives
Derivative — The fundamental operation appearing in every differential equation
First Derivative — The specific derivative order that defines first order equations
Second Order Differential Equation — The next higher order differential equation
Initial Value Problem — A first order equation paired with an initial condition