Nonhomogeneous Differential Equation

Homogeneous vs Nonhomogeneous Differential Equation

From reading about differential equations, you already know that there are several ways to classify them. A useful one is homogeneous versus nonhomogeneous.

For example, the differential equation

\[ y'' + 2y' - 3xy = 0\]

is a homogeneous differential equation. It can be written with all of the functions involving the dependent variable on one side of the equation, and zero on the other side.

By contrast, the differential equation

\[ y'' + 2y' - 3xy = \sin x\]

is a nonhomogeneous differential equation. It has a function of the independent variable, \(x\), on one side of the equation instead of the zero that the previous example had.

It is important to note that whether or not a differential equation is homogeneous or nonhomogeneous has nothing to do with the order of the equation or whether or not it is linear!

What is a Linear Nonhomogeneous Differential Equation?

Now that you know a differential equation can be both linear and nonhomogeneous, doesn't have to be both linear and nonhomogeneous, let's take a look at the case where it is.

Remember the properties of a linear differential equation:

• each dependent variable appears in linear fashion;

• the dependent variable and/or its derivatives are all raised to the power of \(1\);

• none of the dependent variables and/or its derivatives are multiplied together;

• the dependent variable and/or its derivatives can not be part of a special function, like a trigonometric function or the exponential function; and

• the independent variable can be non-linear (raised to a power, part of special function, etc.).

Here are some examples of linear differential equations:

• \( \dfrac{\mathrm{d}y}{\mathrm{d}t}+2y=0\);

• \( t^2\dfrac{\mathrm{d}^2y}{\mathrm{d}t^2} +t\dfrac{\mathrm{d}y}{\mathrm{d}t}+4y=e^t\); and

• \( \dfrac{\mathrm{d}y}{\mathrm{d}t}+\left(\cos{t}\right)y=t^2\).

Of the equations listed above, only

\[ \dfrac{\mathrm{d}y}{\mathrm{d}t}+2y=0\]

is a homogeneous equation. Both of the others are nonhomogeneous. So while a differential equation can be both linear and nonhomogeneous, it doesn't have to be.

How you solve a nonhomogeneous linear equation varies depending on whether or not it is first order. Read on for techniques to solve first order nonhomogeneous differential equations.

Linear Nonhomogeneous Differential Equations with Constant Coefficients

Let's start with the constant coefficient case. A linear first order nonhomogeneous differential equation with constant coefficients has the form

\[ \frac{\mathrm{d}y}{\mathrm{d}x}+ay=f(x),\]

although this is more commonly written as

\[y'+ay=f(x).\]

The idea is to use an integrating factor to solve this kind of separable differential equation. First order linear constant coefficient equations are especially nice since the integrating factor for these sorts of equations is

\[ h(x) = e^{ax}.\]

You would then multiply both sides of the equation by the integrating factor, and integrate giving you

\[ \int (e^{ax}y)'\, \mathrm{d}x = \int e^{ax} f(x) \, \mathrm{d}x ,\]

so

\[ e^{ax}y(x) = \int e^{ax} f(x) \, \mathrm{d}x ,\]

or

\[y(x) = e^{-ax} \int e^{ax} f(x) \, \mathrm{d}x .\]

So being able to find an explicit solution to this type of equation really depends on whether or not you can integrate \(e^{ax} f(x) \). Let's take a look at a quick example.

Next, let's take a look at a more general first order nonhomogeneous equation.

solving First Order Nonhomogeneous Differential Equation

A general linear first order nonhomogeneous differential equation with constant coefficients has the form

\[y'+a(x)y=f(x).\]

This kind of equation is still solved using an integrating factor. Here the integrating factor is

\[ h(x) = e^{\int a(x)\,\mathrm{d} x},\]

and the solution to the differential equation is

\[y(x) = e^{-\int a(x)\,\mathrm{d} x} \int e^{\int a(x)\,\mathrm{d} x} f(x) \, \mathrm{d}x .\]

That means finding an explicit solution still depends on being able to find a closed form integral!

Let's take a look at an example.

More examples are always good!

Nonhomogeneous Linear Equation Examples

Let's take a look at some more examples.

Let's look at another example.