Annihilator method

In mathematics, the annihilator method is a procedure used to find a particular solution to certain types of inhomogeneous ordinary differential equations. It is equivalent to the method of undetermined coefficients, and the two names are sometimes used interchangeably, although they can describe different techniques. The phrase "undetermined coefficients" can also be used to refer to the step in the annihilator method in which the coefficients are calculated.

The annihilator method is used as follows. Given the ODE $P(D)y\; =\; f(x)$, find another differential operator $A(D)$ such that $A(D)f(x)\; =\; 0$. This operator is called the annihilator, thus giving the method its name. Applying $A(D)$ to both sides of the ODE gives a homogeneous ODE for which we find a solution basis $\{y\_1,ldots,y\_n\}$ as before. Then the original inhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combinations to satisfy the ODE.

This method is not as general as variation of parameters in the sense that an annihilator does not always exist.

Example

Given $y"-4y\text{'}+5y=sin(kx)$, $P(D)=D^2-4D+5$.The simplest annihilator of $sin(kx)$ is $A(D)=D^2+k^2$. The zeros of $A(z)P(z)$ are $\{2+i,2-i,ik,-ik\}$, so the solution basis of $A(D)P(D)$ is $\{y\_1,y\_2,y\_3,y\_4\}=\{e^\{(2+i)x\},e^\{(2-i)x\},e^\{ikx\},e^\{-ikx\}\}.$

Setting $y=c\_1y\_1+c\_2y\_2+c\_3y\_3+c\_4y\_4$ we find:giving the system:$i=(k^2+4ik-5)c\_3+(-k^2+4ik+5)c\_4$:$0=(k^2+4ik-5)c\_3+(k^2-4ik-5)c\_4$which has solutions:$c\_3=frac\; i\{2(k^2+4ik-5)\}$, $c\_4=frac\; i\{2(-k^2+4ik+5)\}$giving the solution set:This solution can be broken down into the homogeneous and nonhomogenous parts. In particular, $y\_p\; =\; frac\{4kcos(kx)+(5-k^2)sin(kx)\}\{k^4+6k^2+25\}$ is a particular solution to the nonhomogeneous differential equation, and $y\_c\; =\; c\_1y\_1\; +\; c\_2y\_2$ is a complementary solution to the corresponding homogeneous equation. The values of $c\_1$ and $c\_2$ are determined usually through a set of initial conditions. Since this is a second order equation, there would be two such conditions necessary in order to fully determine these values.

The fundamental solutions $y\_1\; =\; e^\{(2+i)x\}$ and $y\_1\; =\; e^\{(2-i)x\}$ can be further rewritten using Euler's formula:

$e^\{(2+i)x\}\; =\; e^2\; e^\{ix\}\; =\; e^2\; (cos\; x\; +\; i\; sin\; x)$

$e^\{(2-i)x\}\; =\; e^2\; e^\{ix\}\; =\; e^2\; (cos\; x\; -\; i\; sin\; x)$

Then $c\_1\; y\_1\; +\; c\_2\; y\_2\; =\; c\_1e^2\; (cos\; x\; +\; i\; sin\; x)\; +\; c\_2\; e^2\; (cos\; x\; -\; i\; sin\; x)\; =\; (c\_1\; +\; c\_2)\; e^2\; cos\; x\; +\; i(c\_1\; -\; c\_2)\; e^2\; sin\; x$, and a suitable reassignment of the constants gives a simpler and more understandable form of the complementary solution: $y\_c\; =\; e^2\; (c\_1\; cos\; x\; +\; c\_2\; sin\; x)$.