Complementary Solution

Definition

The complementary solution is a particular solution to a nonhomogeneous linear differential equation that, when combined with the general solution of the corresponding homogeneous equation, yields the complete solution to the original nonhomogeneous equation.

5 Must Know Facts For Your Next Test

1. The complementary solution represents the part of the complete solution that is independent of the nonhomogeneous or forcing term in the differential equation.
2. The complementary solution can be found by solving the corresponding homogeneous differential equation, which is the original equation with the nonhomogeneous term set to zero.
3. The form of the complementary solution depends on the roots of the characteristic equation associated with the homogeneous differential equation.
4. If the homogeneous equation has constant coefficients, the complementary solution can be expressed as a linear combination of exponential, trigonometric, or hyperbolic functions.
5. The complementary solution, when combined with the particular solution, gives the complete solution to the original nonhomogeneous differential equation.

Review Questions

• Explain the role of the complementary solution in the context of nonhomogeneous linear differential equations.
  • The complementary solution represents the part of the complete solution to a nonhomogeneous linear differential equation that is independent of the nonhomogeneous or forcing term. It is obtained by solving the corresponding homogeneous equation, which is the original equation with the nonhomogeneous term set to zero. The complementary solution, when combined with the particular solution, yields the complete solution to the original nonhomogeneous differential equation.
• Describe how the form of the complementary solution is determined based on the roots of the characteristic equation associated with the homogeneous differential equation.
  • The form of the complementary solution depends on the roots of the characteristic equation associated with the homogeneous differential equation. If the homogeneous equation has constant coefficients, the complementary solution can be expressed as a linear combination of exponential, trigonometric, or hyperbolic functions, depending on the nature of the roots. Real roots lead to exponential functions, complex conjugate roots lead to trigonometric functions, and repeated roots can result in the inclusion of polynomials multiplied by the exponential or trigonometric terms.
• Analyze the relationship between the complementary solution and the general solution to a nonhomogeneous linear differential equation.
  • The complementary solution, when combined with the particular solution, gives the complete or general solution to the original nonhomogeneous differential equation. The general solution includes all possible solutions to the equation and can be expressed as the sum of the complementary solution, which represents the part of the solution that is independent of the nonhomogeneous term, and the particular solution, which satisfies the specific nonhomogeneous equation. Understanding the relationship between the complementary solution and the general solution is crucial in solving nonhomogeneous linear differential equations and applying the appropriate techniques to find the complete solution.

Related terms

Homogeneous Differential Equation: A homogeneous differential equation is a differential equation where the right-hand side is equal to zero, meaning there is no nonhomogeneous or forcing term present.

Particular Solution:

The particular solution is a specific solution to a nonhomogeneous linear differential equation that satisfies the equation, but does not necessarily satisfy the initial or boundary conditions.

General Solution:

The general solution to a linear differential equation is the set of all solutions to the equation, and includes the particular solution as well as the complementary solution.