 # Homogeneous Differential Equation – Definition, Solutions, and Examples

Understanding how to work with homogeneous differential equations is important if we want to explore more complex calculus topics and work on advanced endeavors in other disciplines such as physics, mathematics, and finance.

Homogeneous differential equations are differential equations where each term will be of the form, $y^{(n)}q(x)$. Oftentimes, you’ll see homogeneous equations having zero on the right-hand side of their equation.

This article gives you a comprehensive idea of what makes homogeneous differential equations unique. We’ll also summarize the important techniques you’ll need to work on different types of homogeneous differential equations. For now, let’s begin by making sure that we know how to identify and rewrite homogeneous differential equations.

## What Is a Homogeneous Differential Equation?

A homogeneous differential equation contains a differential expression on its left-hand side and zero on the right-hand side of the equation. Using this definition, the general form of a homogeneous differential equation is as shown below.

\begin{aligned}a_n(x) y^{(n)} + a_{n -1}(x) y^{(n – 1)} +…+a_1y^{\prime}+ a_0y &= 0\end{aligned}

In this general form, $a_n(x), a_{n -1}(x), …, a_1(x), a_0(x)$, are factors in terms of $x$. When the homogeneous differential equation is also linear, these become constant factors, $a_n, a_{n-1}, …,a_1, a_0$. In the past, we’ve also learned about first order and second order differential equations. Below are their general forms as we have learned before:

 First Order Homogeneous Differential Equation \begin{aligned}\dfrac{dy}{dx} &= f(x, y)\\ P(x, y) \phantom{x}dx + Q(x, y) \phantom{x}dy &= 0\end{aligned} Second Order Homogeneous Differential Equation \begin{aligned}y^{\prime \prime} + P(x)y^{\prime} + Q(x)y &= f(x)\\\dfrac{d^2y}{dx^2}+ P(x)\dfrac{dy}{dx} + Q(x)y &= f(x) \end{aligned}

Use these definitions and the general form of the homogeneous differential equations to classify them. Knowing how to identify homogeneous differential equations is important since techniques for solving differential equations would depend on whether they are homogeneous or not.

### How To Tell if a Differential Equation Is Homogeneous

Suppose that we have a differential equation, $a_n(x) y^{(n)} + a_{n -1}(x) y^{(n – 1)} +…+a_1y^{\prime}+ a_0y =g(x)$, we can identify whether it is homogeneous or not depending on the value of $g(x)$.

• When we can show that $g(x) = 0$, the differential equation is homogeneous.
• Otherwise, the equation is non-homogeneous.

Let us show you two examples to demonstrate how a differential equation looks when it is homogeneous and when it is non-homogeneous.

\begin{aligned}\boldsymbol{y^{\prime \prime} \cos x – y\sin  x = y^{\prime}}\end{aligned}

We can rewrite the equation so that all terms with $y$ and its derivatives are on the left-hand side.

\begin{aligned} y^{\prime \prime} \cos x – y\sin  x &= y^{\prime}\\ y^{\prime \prime} \cos x -y^{\prime} – y\sin  x &=0 \end{aligned}

Since we’ve written the differential equation in its standard form, we have shown that $g(x) = 0$, so our equation is indeed homogeneous. Now, let’s show you an example of a non-homogeneous equation for comparison.

\begin{aligned}\boldsymbol{y^{\prime \prime} \cos x +  y\cos  x – 5xe^x = 0}\end{aligned}

Once again, we rewrite the differential equation in the form of $a_n(x) y^{(n)} + a_{n -1}(x) y^{(n – 1)} +…+a_1y^{\prime}+ a_0y =g(x)$ and see if $g(x)$ is zero or not.

\begin{aligned}y^{\prime \prime} \cos x +  y\cos  x – 5xe^x &= 0\\ y^{\prime \prime} \cos x +  y\cos  x &= 5xe^x \end{aligned}

Since $g(x) = 5xe^x$ and it’s not equal to zero, the equation is not homogeneous, or a non-homogeneous differential equation. Now that we know how to identify homogeneous equations, it’s time for us to refresh our knowledge on the different techniques of finding the solutions of homogeneous differential equations.

## How To Solve Homogeneous Differential Equations?

There are different ways for us to solve homogeneous differential equations and it’s important that we know the common techniques used when dealing with them. First, let’s lay out the methods we have for first order homogeneous differential equations.