Higher-order linear differential equations often include nonhomogeneous terms, making them trickier to solve. This section introduces methods for tackling these equations, focusing on the method of undetermined coefficients for finding particular solutions.

Understanding nonhomogeneous equations is crucial for modeling real-world systems with external inputs. We'll learn how to break down solutions into complementary and particular parts, and use clever guessing to find particular solutions for common forcing functions.

Nonhomogeneous Equations

Solving Nonhomogeneous Equations

• Nonhomogeneous linear differential equations contain a term that is a function of the independent variable, often called the forcing function $f(t)$
• The general solution to a nonhomogeneous equation consists of the sum of a particular solution and the complementary function
  • Particular solution $y_p(t)$ is any solution that satisfies the nonhomogeneous equation
  • Complementary function $y_c(t)$ is the general solution to the corresponding homogeneous equation obtained by setting $f(t) = 0$
• Superposition principle for nonhomogeneous equations states that the general solution is the sum of the particular solution and the complementary function: $y(t) = y_p(t) + y_c(t)$
  • Allows for solving nonhomogeneous equations by finding $y_p(t)$ and $y_c(t)$ separately, then adding them together

Forcing Function and Its Impact

• The forcing function $f(t)$ is the nonhomogeneous term in the differential equation that makes it nonhomogeneous
  • Represents an external input or forcing term that drives the system
  • Can be a function of the independent variable $t$, such as $\sin(t)$, $e^{-t}$, or a polynomial
• The particular solution $y_p(t)$ depends on the form of the forcing function $f(t)$
  • Different forcing functions lead to different particular solutions
  • The choice of the method for finding the particular solution (such as undetermined coefficients or variation of parameters) depends on the form of $f(t)$

Method of Undetermined Coefficients

Overview and Trial Functions

• The method of undetermined coefficients is a technique for finding a particular solution to a nonhomogeneous linear differential equation
  • Applicable when the forcing function $f(t)$ is a polynomial, exponential, sine, cosine, or a combination of these
• The method involves assuming a trial function $y_p(t)$ with unknown coefficients based on the form of $f(t)$
  • For a polynomial forcing function of degree $n$, the trial function is a polynomial of degree $n$ with undetermined coefficients
  • For an exponential forcing function $e^{at}$, the trial function is $Ae^{at}$ with an undetermined coefficient $A$
  • For sine or cosine forcing functions, the trial function includes both sine and cosine terms with undetermined coefficients

Annihilator Method and Resonance

• The annihilator method is a variation of the method of undetermined coefficients that helps determine the appropriate form of the trial function
  • Involves applying a differential operator (annihilator) that eliminates the forcing function $f(t)$
  • The trial function is modified based on the annihilator to ensure that it does not get eliminated when the annihilator is applied
• Resonance occurs when the forcing function is a solution to the corresponding homogeneous equation
  • In this case, the trial function needs to be multiplied by $t$ to avoid duplication with the complementary function
  • For example, if $f(t) = e^{-t}$ and the complementary function includes $e^{-t}$, the trial function should be $Ate^{-t}$ instead of $Ae^{-t}$