Annihilator method
The annihilator method is a systematic technique for determining particular solutions to nonhomogeneous linear ordinary differential equations with constant coefficients, particularly when the nonhomogeneous term consists of polynomials, exponentials, trigonometric functions, or their products. It leverages differential operators—such as powers of the derivative D = \frac{d}{dx}—that reduce the specified forcing function to zero, thereby transforming the original nonhomogeneous equation into a higher-order homogeneous one whose general solution encompasses the desired particular solution.[1] This approach builds on the method of undetermined coefficients by providing a structured way to guess the form of the particular solution without trial and error for the coefficients alone. An annihilator is defined as a linear differential operator A(D) such that A(D)[g(x)] = 0, where g(x) is the nonhomogeneous term; for instance, D^{n+1} annihilates any polynomial of degree n, (D - \alpha) annihilates e^{\alpha x}, and (D^2 + \omega^2) annihilates \sin(\omega x) or \cos(\omega x). For composite functions, the annihilator is the product of individual operators, ensuring the method applies efficiently to common forcing functions encountered in applications like physics and engineering.[2] The procedure typically begins by solving the associated homogeneous equation to obtain the complementary solution y_c. An appropriate annihilator is then identified and applied to both sides of the original equation L = g(x), yielding A(L) = 0, a new homogeneous equation. The general solution to this elevated-order equation is found, and the components overlapping with y_c are discarded to isolate the particular solution y_p. Finally, the general solution is y = y_c + y_p, with constants determined via initial or boundary conditions if provided. This method's advantages include its algorithmic nature and avoidance of integration, making it preferable over variation of parameters for suitable g(x).[1]Background
Linear Differential Equations with Constant Coefficients
Linear ordinary differential equations (ODEs) with constant coefficients form a fundamental class in the study of differential equations, characterized by their linear structure and unchanging coefficients. The general form of an nth-order linear ODE with constant coefficients is given by a_n y^{(n)}(x) + a_{n-1} y^{(n-1)}(x) + \dots + a_1 y'(x) + a_0 y(x) = g(x), where a_n, a_{n-1}, \dots, a_0 are constants with a_n \neq 0, y^{(k)}(x) denotes the kth derivative of y(x), and g(x) represents the nonhomogeneous forcing term.[3] This equation models numerous physical phenomena, such as vibrations and electrical circuits, due to the tractability provided by the constant coefficients.[4] For the homogeneous case, where g(x) = 0, the equation simplifies to a_n y^{(n)}(x) + a_{n-1} y^{(n-1)}(x) + \dots + a_0 y(x) = 0. The solution method relies on the characteristic equation, obtained by assuming a solution of the form y(x) = e^{rx} and substituting it into the homogeneous equation, yielding the algebraic equation a_n r^n + a_{n-1} r^{n-1} + \dots + a_1 r + a_0 = 0. This polynomial equation in r has n roots (counting multiplicities), which determine the form of the general solution.[5][6] The roots of the characteristic equation dictate the basis functions for the general solution. For a distinct real root r_k, the corresponding solution component is c_k e^{r_k x}, where c_k is an arbitrary constant. If a real root r has multiplicity m, the solutions include e^{rx}, x e^{rx}, \dots, x^{m-1} e^{rx}. For complex conjugate roots \alpha \pm i\beta (with \beta \neq 0) of multiplicity one, the real-valued solutions are e^{\alpha x} \cos(\beta x) and e^{\alpha x} \sin(\beta x); for multiplicity m, the set extends to include factors of x^j for j = 0 to m-1 multiplied by these trigonometric functions. The general solution is a linear combination of these n linearly independent basis functions, ensuring a complete n-dimensional solution space.[5][7] To facilitate analysis, these equations can be expressed using the differential operator D = \frac{d}{dx}, where higher powers represent higher derivatives. The homogeneous equation then becomes p(D) y = 0, with p(D) = a_n D^n + a_{n-1} D^{n-1} + \dots + a_0 being the characteristic polynomial evaluated at the operator D. This operator notation highlights the linearity and commutativity of derivatives, aiding in factorization and solution construction.[5][8]Homogeneous and Nonhomogeneous Equations
A homogeneous linear differential equation with constant coefficients is expressed in operator notation as p(D) y = 0, where D = \frac{d}{dx} denotes the differentiation operator and p(D) = a_n D^n + a_{n-1} D^{n-1} + \cdots + a_1 D + a_0 is a polynomial of degree n with constant coefficients a_i (and a_n \neq 0).[9] The general solution to this equation consists of a linear combination of n linearly independent fundamental solutions: y_h(x) = c_1 y_1(x) + c_2 y_2(x) + \cdots + c_n y_n(x), where the c_i are arbitrary constants determined by initial or boundary conditions.[10] This structure arises from the fact that the solution space forms an n-dimensional vector space under addition and scalar multiplication, as guaranteed by the linearity of the operator.[11] In contrast, a nonhomogeneous linear differential equation with constant coefficients has the form p(D) y = g(x), where g(x) is a given non-zero forcing function (also called the nonhomogeneous term).[12] The general solution to this equation is the sum of the homogeneous solution and a particular solution: y(x) = y_h(x) + y_p(x), where y_p(x) is any specific function satisfying p(D) y_p = g(x).[10] The arbitrary constants in y_h(x) allow adjustment to meet initial conditions, while y_p(x) accounts for the influence of g(x). Solutions to the associated homogeneous equation fail to satisfy the nonhomogeneous equation because p(D) y_h = 0 \neq g(x) for g(x) \not\equiv 0.[11] However, the linearity of p(D) ensures the superposition principle applies: if y_h solves the homogeneous equation and y_p solves the nonhomogeneous one, then p(D)(y_h + y_p) = p(D) y_h + p(D) y_p = 0 + g(x) = g(x), so their sum solves the full equation.[11] This principle underpins the decomposition of the solution and extends to linear combinations of multiple particular solutions when g(x) is a sum of terms. The foundational work on homogeneous linear differential equations with constant coefficients dates to the 18th century, with key developments by Leonhard Euler and Joseph-Louis Lagrange. Euler established a general solution method using exponential functions, as detailed in his correspondence around 1743.[13]The Annihilator Method
Definition and Principle
The annihilator method provides a systematic approach to finding particular solutions for nonhomogeneous linear ordinary differential equations with constant coefficients by leveraging differential operators. Central to this method is the concept of an annihilator, defined as a linear differential operator A(D), where D = \frac{d}{dx} denotes the differentiation operator, such that A(D) g(x) = 0 for the given nonhomogeneous term g(x). This operator effectively "annihilates" the forcing function, transforming it into the zero function.[1] The underlying principle exploits the properties of operator composition. Consider the original nonhomogeneous equation p(D) y = g(x), where p(D) is the characteristic operator of the homogeneous part. Since A(D) g(x) = 0, applying A(D) to both sides yields A(D) p(D) y = 0, converting the problem into a higher-order homogeneous equation whose general solution encompasses both the solution to the original homogeneous equation p(D) y = 0 and the desired particular solution y_p. This transformation allows the particular solution to be extracted from the broader solution space.[1] Within the general solution to A(D) p(D) y = 0, the particular solution y_p consists of those components that do not belong to the solution space of the original homogeneous equation p(D) y = 0. Any terms overlapping with the homogeneous solution are discarded to isolate y_p, ensuring it satisfies the nonhomogeneous equation without redundancy.[1] A key aspect of the method is the selection of a minimal annihilator, which is the lowest-degree operator that satisfies A(D) g(x) = 0. Higher-degree annihilators may also work but introduce unnecessary complexity by expanding the order of the transformed equation beyond what is required.[1]Annihilators for Common Forcing Functions
The annihilator method identifies linear differential operators with constant coefficients that, when applied to a forcing function g(x), yield zero. These operators are essential for transforming nonhomogeneous equations into homogeneous ones. The following table summarizes annihilators for standard classes of forcing functions encountered in linear differential equations with constant coefficients.[2]| Forcing Function g(x) | Annihilator |
|---|---|
| Polynomial: x^m | D^{m+1} |
| Exponential: e^{ax} | D - a |
| Trigonometric: \sin(bx) or \cos(bx) | D^2 + b^2 |
| Product: x^m e^{ax} | (D - a)^{m+1} |
| Product: x^m e^{ax} \sin(bx) or x^m e^{ax} \cos(bx) | ((D - a)^2 + b^2)^{m+1} |
Step-by-Step Procedure
The annihilator method provides a systematic approach to finding a particular solution y_p for a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients, of the form p(D) y = g(x), where p(D) is a polynomial differential operator and g(x) is the forcing function. This technique leverages the fact that applying an annihilator operator A(D) to both sides of the equation yields a higher-order homogeneous equation whose general solution encompasses both the original homogeneous solution y_h and the particular solution y_p, along with extraneous terms. The method is particularly effective for forcing functions that are polynomials, exponentials, sines, cosines, or their products, as these admit simple annihilators.[1][14] To apply the method, follow these steps:- Express the given nonhomogeneous ODE in operator form as p(D) y = g(x), where p(D) is the characteristic polynomial operator corresponding to the left-hand side, and identify the minimal annihilator A(D) such that A(D) g(x) = 0. The minimal annihilator is the lowest-order differential operator that reduces g(x) and all its derivatives to zero; for common forms like polynomials or trigonometric functions, these are tabulated in standard references.[1][15]
- Apply the annihilator to both sides of the original equation to obtain the higher-order homogeneous equation A(D) p(D) y = 0. Solve this auxiliary equation by finding its characteristic roots and writing the general solution as y = y_c, which decomposes into the original homogeneous solution y_h (from p(D) y = 0), the particular solution y_p, and extraneous terms y_{extra} arising from the additional factors in A(D). The form of y_c is a linear combination of basis functions like e^{rx}, x^k e^{rx}, e^{\alpha x} \cos(\beta x), and e^{\alpha x} \sin(\beta x), determined by the roots of the characteristic equation for A(D) p(D).[14][1]
- From the general solution y_c of the auxiliary equation, isolate y_p by discarding all terms that belong to the original homogeneous solution y_h. This ensures y_p is free of arbitrary constants already accounted for in y_h; the remaining terms, including any multiplicity adjustments from step 4, form y_p. If no overlap occurs, y_p consists directly of the non-homogeneous components introduced by A(D).[15][1]
- Verify that the isolated y_p satisfies the original nonhomogeneous ODE by substituting it back into p(D) y = g(x) and confirming equality. The full solution is then y = y_h + y_p, where constants in y_h are determined by initial or boundary conditions if provided.[14][1]