In mathematics , the annihilator method is a procedure used to find a particular solution to certain types of non-homogeneous ordinary differential equations (ODEs). It is similar to the method of undetermined coefficients , but instead of guessing the particular solution in the method of undetermined coefficients , the particular solution is determined systematically in this technique. The phrase undetermined coefficients can also be used to refer to the step in the annihilator method in which the coefficients are calculated.
17-581: [REDACTED] Look up annihilator in Wiktionary, the free dictionary. Annihilator(s) may refer to: Mathematics [ edit ] Annihilator (ring theory) Annihilator (linear algebra) , the annihilator of a subset of a vector subspace Annihilator method , a type of differential operator, used in a particular method for solving differential equations Annihilator matrix , in regression analysis Music [ edit ] Annihilator (band) ,
34-402: A 1986 television film starring Mark Lindsay Chapman The Annihilators (film) , a 1985 action film by Charles E. Sellier Jr. The Annihilators (novel) , a 1983 novel by Donald Hamilton See also [ edit ] Annihilation (disambiguation) Annihilating element Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
51-402: A 1986 television film starring Mark Lindsay Chapman The Annihilators (film) , a 1985 action film by Charles E. Sellier Jr. The Annihilators (novel) , a 1983 novel by Donald Hamilton See also [ edit ] Annihilation (disambiguation) Annihilating element Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with
68-404: A Canadian heavy metal band Annihilator (album) , a 2010 album by the aforementioned band Other media [ edit ] Annihilator (Justice League) , an automaton in the fictional series Justice League Unlimited Annihilators (Marvel Comics) , a team of superheroes Annihilator , a 2015 science fiction comic by Grant Morrison and Frazer Irving Annihilator (film) ,
85-404: A Canadian heavy metal band Annihilator (album) , a 2010 album by the aforementioned band Other media [ edit ] Annihilator (Justice League) , an automaton in the fictional series Justice League Unlimited Annihilators (Marvel Comics) , a team of superheroes Annihilator , a 2015 science fiction comic by Grant Morrison and Frazer Irving Annihilator (film) ,
102-1183: A set of initial conditions. Since this is a second-order equation, two such conditions are necessary to determine these values. The fundamental solutions y 1 = e ( 2 + i ) x {\displaystyle y_{1}=e^{(2+i)x}} and y 2 = e ( 2 − i ) x {\displaystyle y_{2}=e^{(2-i)x}} can be further rewritten using Euler's formula : Then c 1 y 1 + c 2 y 2 = c 1 e 2 x ( cos x + i sin x ) + c 2 e 2 x ( cos x − i sin x ) = ( c 1 + c 2 ) e 2 x cos x + i ( c 1 − c 2 ) e 2 x sin x {\displaystyle c_{1}y_{1}+c_{2}y_{2}=c_{1}e^{2x}(\cos x+i\sin x)+c_{2}e^{2x}(\cos x-i\sin x)=(c_{1}+c_{2})e^{2x}\cos x+i(c_{1}-c_{2})e^{2x}\sin x} , and
119-490: Is A ( D ) = D 2 + k 2 {\displaystyle A(D)=D^{2}+k^{2}} . The zeros of A ( z ) P ( z ) {\displaystyle A(z)P(z)} are { 2 + i , 2 − i , i k , − i k } {\displaystyle \{2+i,2-i,ik,-ik\}} , so the solution basis of A ( D ) P ( D ) {\displaystyle A(D)P(D)}
136-689: Is { y 1 , y 2 , y 3 , y 4 } = { e ( 2 + i ) x , e ( 2 − i ) x , e i k x , e − i k x } . {\displaystyle \{y_{1},y_{2},y_{3},y_{4}\}=\{e^{(2+i)x},e^{(2-i)x},e^{ikx},e^{-ikx}\}.} Setting y = c 1 y 1 + c 2 y 2 + c 3 y 3 + c 4 y 4 {\displaystyle y=c_{1}y_{1}+c_{2}y_{2}+c_{3}y_{3}+c_{4}y_{4}} we find giving
153-489: Is a particular integral for the nonhomogeneous differential equation, and y c = c 1 y 1 + c 2 y 2 {\displaystyle y_{c}=c_{1}y_{1}+c_{2}y_{2}} is a complementary solution to the corresponding homogeneous equation. The values of c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} are determined usually through
170-454: Is used as follows. Given the ODE P ( D ) y = f ( x ) {\displaystyle P(D)y=f(x)} , find another differential operator A ( D ) {\displaystyle A(D)} such that A ( D ) f ( x ) = 0 {\displaystyle A(D)f(x)=0} . This operator is called the annihilator , hence the name of
187-641: The real numbers . If f ( x ) {\displaystyle f(x)} consists of the sum of the expressions given in the table, the annihilator is the product of the corresponding annihilators. Given y ″ − 4 y ′ + 5 y = sin ( k x ) {\displaystyle y''-4y'+5y=\sin(kx)} , P ( D ) = D 2 − 4 D + 5 {\displaystyle P(D)=D^{2}-4D+5} . The simplest annihilator of sin ( k x ) {\displaystyle \sin(kx)}
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#1732855292222204-512: The 💕 [REDACTED] Look up annihilator in Wiktionary, the free dictionary. Annihilator(s) may refer to: Mathematics [ edit ] Annihilator (ring theory) Annihilator (linear algebra) , the annihilator of a subset of a vector subspace Annihilator method , a type of differential operator, used in a particular method for solving differential equations Annihilator matrix , in regression analysis Music [ edit ] Annihilator (band) ,
221-464: The method. Applying A ( D ) {\displaystyle A(D)} to both sides of the ODE gives a homogeneous ODE ( A ( D ) P ( D ) ) y = 0 {\displaystyle {\big (}A(D)P(D){\big )}y=0} for which we find a solution basis { y 1 , … , y n } {\displaystyle \{y_{1},\ldots ,y_{n}\}} as before. Then
238-513: The original inhomogeneous ODE is used to construct a system of equations restricting the coefficients of the linear combination to satisfy the ODE. This method is not as general as variation of parameters in the sense that an annihilator does not always exist. Where k {\displaystyle k} is in the natural numbers , and k , b , a , c 1 , ⋯ , c k {\displaystyle k,b,a,c_{1},\cdots ,c_{k}} are in
255-486: The system which has solutions giving the solution set This solution can be broken down into the homogeneous and nonhomogeneous parts. In particular, y p = 4 k cos ( k x ) + ( 5 − k 2 ) sin ( k x ) k 4 + 6 k 2 + 25 {\displaystyle y_{p}={\frac {4k\cos(kx)+(5-k^{2})\sin(kx)}{k^{4}+6k^{2}+25}}}
272-467: The title Annihilator . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Annihilator&oldid=1247377402 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages annihilator From Misplaced Pages,
289-480: The title Annihilator . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Annihilator&oldid=1247377402 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Annihilator method The annihilator method
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