There are two definitions of the term “homogeneous differential equation.” One definition calls a first-order equation of the form

homogeneous if M and N are both homogeneous functions of the same degree. The second definition — and the one which you'll see much more often—states that a differential equation (of any order) is homogeneous if once all the terms involving the unknown function are collected together on one side of the equation, the other side is identically zero. For example,

but

The nonhomogeneous equation

can be turned into a homogeneous one simply by replacing the right-hand side by 0:

Equation (**) is called the homogeneous equation corresponding to the nonhomogeneous equation, (*). There is an important connection between the solution of a nonhomogeneous linear equation and the solution of its corresponding homogeneous equation. The two principal results of this relationship are as follows:

Theorem A. If y₁( x) and y₂( x) are linearly independent solutions of the linear homogeneous equation (**), then every solution is a linear combination of y₁ and y₂. That is, the general solution of the linear homogeneous equation is

Theorem B. If y( x) is any particular solution of the linear nonhomogeneous equation (*), and if y _h ( x) is the general solution of the corresponding homogeneous equation, then the general solution of the linear nonhomogeneous equation is

That is,

[Note: The general solution of the corresponding homogeneous equation, which has been denoted here by y _h , is sometimes called the complementary function of the nonhomogeneous equation (*).] Theorem A can be generalized to homogeneous linear equations of any order, while Theorem B as written holds true for linear equations of any order. Theorems A and B are perhaps the most important theoretical facts about linear differential equations—definitely worth memorizing.

Example 1: The differential equation

is satisfied by the functions

Verify that any linear combination of y₁ and y₂ is also a solution of this equation. What is its general solution?

Every linear combination of y₁ = e ^x and y₂ = xe ^x looks like this:

for some constants c₁ and c₂. To verify that this satisfies the differential equation, just substitute. If y = c₁ e ^x + c₂ xe ^x , then

Substituting these expressions into the left-hand side of the given differential equation gives

Thus, any linear combination of y₁ = e ^x and y₂ = xe ^x does indeed satisfy the differential equation. Now, since y₁ = e ^x and y₂ = xe ^x are linearly independent, Theorem A says that the general solution of the equation is

Example 2: Verify that y = 4 x – 5 satisfies the equation

Then, given that y₁ = e^{− x} and y₂ = e^{− 4x} are solutions of the corresponding homogeneous equation, write the general solution of the given nonhomogeneous equation.

First, to verify that y = 4 x – 5 is a particular solution of the nonhomogeneous equation, just substitute. If y = 4 x – 5, then y′ = 4 and y″ = 0, so the left-hand side of the equation becomes

Now, since the functions y₁ = e^{− x} and y₂ = e^{− 4x} are linearly independent (because neither is a constant multiple of the other), Theorem A says that the general solution of the corresponding homogeneous equation is

Theorem B then says

is the general solution of the given nonhomogeneous equation.

Example 3: Verify that both y₁ = sin x and y₂ = cos x satisfy the homogeneous differential equation y″ + y = 0. What then is the general solution of the nonhomogeneous equation y″ + y = x?

If y₁ = sin x, then y″₁ + y₁ does indeed equal zero. Similarly, if y₂ = cos x, then y″₂ = y is also zero, as desired. Since y₁ = sin x and y₂ = cos x are linearly independent, Theorem A says that the general solution of the homogeneous equation y″ + y = 0 is

Now, to solve the given nonhomogeneous equation, all that is needed is any particular solution. By inspection, you can see that y = x satisfies y″ + y = x. Therefore, according to Theorem B, the general solution of this nonhomogeneous equation is

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