Solving Second Order (and higher) Non-Homogeneous Differential Equations Using the Annihilator Method


Consider a 2nd order non-homogeneous differential equation of the form:

$$ay''+by'+cy=g(x) $$

We will first solve the above equation as if it were homogeneous and of the form:

$$ay''+by'+cy=0 \qquad(1)$$

Solving equation (1) is done using the methods described in the Homogeneous 2nd order pages.

After solving for the homogeneous case, we have a general solution (for the homogeneous case) of:

$$y_h = C_1 y_1 + C_2 Y_2 \qquad (2)$$

In the process of obtaining the above general solution, our "characteristic equation" also had two roots where roots r1 and r2 are some numbers m1 and m2 respectively:

$$r_1 = m_1 \;,\; r_2 = m_2 $$

Our original non-homogeneous differential equation can now be written as:



$$y = \frac{1}{a}(D-m_1)^{-1} (D-m_2)^{-1} g(x) \qquad(3)$$

note: The proof of the above expression is not included here

We will also make use of the following theorem (also not proven here) in order to solve for y:

$$(D-m)^{-1} g(x) = e^{mx} \int{}{}e^{-mx}g(x)dx \qquad(4)$$

Theorem #4 will be used in a reiterative fashion until all of the D's are removed from equation (3) and we are left with a particular solution for y (designated as yp)

At this point we will have the general solution to our original non-homogeneous differential equation (designated as yg) that is the sum of our solution to the homogeneous case (eqn #2) and the particular solution (yp)

$$y_g = C_1 y_1 + C_2 Y_2 + y_p$$

In the next page we will take a look at an example

Continue on to Annihilator example (2nd Order)