Topic: About the method of undetermined coefficients & method of variation of parameters. Discuss & compare the advantages and disadvantages of each method.
Illustrate your findings with examples.
DOA: 27th Aug, 2010
DOS: 11th Nov,2010
Submitted to: Submitted by:
Ms. Manreet Shingh . Mr. Anirban Sarkar
Deptt. Of Mathematics Roll.No.RG6005A01 Reg.No, 11006396 Sec. G6005 …
ACKNOWLEDGEMENT
I am very pleased to complete this term project “About the method of undetermined coefficients & method of variation of parameters. Discuss & compare the advantages and disadvantages of each method.
Illustrate your findings with examples.
”
I am thankful to Ms Manreet Singh who guided me during the difficult moments ,I faced during the completion of this project & I am to thankful to librarian sir.
Yours faithfully, Anirban Sarkar.
Abstract,
Method of undetermined coefficients, otherwise known as the Lucky Guess Method, is an approach to finding a particular solution to certain inhomogeneous ordinary and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, a "guess" is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time consuming to perform. Variation of parameters, also known as variation of constants, is a general method to solve inhomogeneous linear ordinary differential equations. It was developed by LaGrange. For first-order inhomogeneous linear differential equations it's usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods are rather heuristics that involve guessing and don't work for all inhomogeneous linear differential equations.
Variation of parameters extends to linear partial differential equations as well, specifically to inhomogeneous problems for linear evolution equations like the heat equation, wave equation, and vibrating plate equation. In this setting, the method is more often known as Duhamel's principle, named for Jean-Marie Duhamel who first applied the method to solve the inhomogeneous heat equation. Sometimes variation of parameters itself is called Duhamel's principle and vice-versa.
Table of content,
1. Introduction, 2. What is the Method of undetermined coefficients & variation of parameters? 3. Advantage of the method of undetermined Coefficients, 4. Disadvantage of the method of undetermined Coefficients, 5. Advantage of method of variation of parameters, 6. Disadvantage of method of variation of parameters, 7. Examples, 8. Applications, 9. Conclusion, 10. Reference.
Introduction,
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modelled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton's laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.
This two methods is also a differential equation.
What is the method of the undetermined coefficients & variation of parameters?
undetermined coefficients
Considering a linear non-homogeneous ordinary differential equation with the form
The method consists of finding the homogeneous solution yc for the complementary function and a particular solution yp based on g(x). Then the general solution y to the equation would be
y = yc + yp
If g(x) consists of the sum of two functions h(x) + w(x) and we say that yp1 is the solution based on h(x) and yp2 the solution based on w(x). Then, using the superposition principle, we can say that the particular solution yp is
yp = yp1 + yp2
variation of parameters
Given an ordinary non-homogeneous linear differential equation of order n
[pic] (i) let [pic] be a fundamental system of the corresponding homogeneous equation
[pic] (ii) Then a particular solution to the non-homogeneous equation is given by
[pic] (iii) where the ci(x) are continuous functions which satisfy the equations
[pic] (iv) (results from substitution of (iii) into the homogeneous case (ii); )
and
[pic] (v) (results from substitution of (iii) into (i) and applying (iv); ci'(x) = 0 for all x and i is the only way to satisfy the condition, since all yi(x) are linearly independent. It implies that all ci(x) are independent of x in the homogeneous case b(x)=0. )
This linear system of n equations can then be solved using Cramer's rule yielding
[pic] where W(x) is the Wronskian determinant of the fundamental system and Wi(x) is the Wronskian determinant of the fundamental system with the i-th column replaced by [pic]
The particular solution to the non-homogeneous equation can then be written as
[pic]
Advantage of the method of undetermined Coefficients,
Equations of the form ay00 + by0 + cy = g(x), for real constants a, b, c and some function g(x). We need to find a SINGLE solution yp to this equation. If we can do this, then the general solution is Ay1+By2 +yp, where Ay1 +By2 is the general solution to the associated homogeneous equation ay00 + by0 + cy = 0. We have discussed two ways of finding a particular solution, undetermined coefficients and variation of parameters.
Method of undetermined coefficients usually faster than variation of parameters, doesn’t require integra-tion.
Disadvantage of the method of undetermined Coefficients,
lots of rules to remember, only works in special cases
The method of undetermined coefficients is essentially the method of solving differ- entail equations by educated guessing. We look at the form of a differential equa- tion, guess what a solution might look like (e.g. Asin x + B cos x or (Ax + B)ex) and then solve for the unknown coefficients. An example: y00 + y = sin 2x
According to the rules set down below, we should guess a particular solution of the form yp = Asin 2x+B cos 2x. We then calculate y00 p +yp = −4Asin 2x−4B cos 2x+
Asin 2x+B cos 2x. We want this to equal sin 2x to get a particular solution. Rear- ranging terms a bit, we get −3Asin 2x − 3B cos 2x = sin 2x. As was the case with polynomials, the coefficients of sin 2x and cos 2x must be the same on both sides of the equation, so we must have −3A = 1 and 3B = 0, i.e., A = −1/3, B = 0. So a particular solution of the above equation is yp = −1/3 sin 2x.
Given a nonhomogeneous equation ay00 + by0 + cy = g(x), here are the rules for the solution type one should guess:
• g(x) = sin kx: guess yp = Asin kx + B cos kx. Use the same guess if g(x) = cos kx or any linear combination of sin kx and cos kx.
• g(x) = ekx: guess yp = Aekx.
• g(x) = P(x)ekx, where P(x) is a polynomial of degree n: guess yp(x) =
Q(x)ekx, where Q(x) is a polynomial of degree n.
• g(x) = ekxP(x) cosmx or g(x) = ekxP(x) sinmx: guess yp(x) = ekxQ(x) cosmx+ ekxR(x) sinmx.
There is one additional thing to remember: if ANY piece of your guessed solution
(i.e., Axe−x or B sin kx) is a solution to the homogeneous equation ay00+by0+cy = 0, multiply your ENTIRE guessed solution by x. Repeat this process if necessary.
Advantage of method of variation of parameters,
a. May be applied to solve any problem.
b. Easy to remember: the only formulae you need are are ªu0 = g and xp = ªu.
Disadvantage of method of variation of parameters,
a. Solving ªu0 = g can get ugly because each component of the vectors involved can contain a complicated expression and row-reduction becomes cumbersome. b. Integrating u0 to find u can sometimes be difficult. c. Doing the matrix multiplication ªu can be tedious.
Examples,
undetermined coefficients
Example 1
Find a particular solution of the equation
[pic] The right side t cos t has the form
[pic] with n=1, α=0, and β=1.
Since α + iβ = i is a simple root of the characteristic equation
[pic] we should try a particular solution of the form
[pic] Substituting yp into the differential equation, we have the identity
[pic] Comparing both sides, we have
[pic]
which has the solution A0 = 0, A1 = 1/4, B0 = 1/4, B1 = 0. We then have a particular solution
[pic]
Example 2
Consider the following linear inhomogeneous differential equation:
[pic] This is like the first example above, except that the inhomogeneous part (ex) is not linearly independent to the general solution of the homogeneous part (c1ex); as a result, we have to multiply our guess by a sufficiently large power of x to make it linearly independent.
Here our guess becomes:
yp = Axex. By substituting this function and its derivative into the differential equation, one can solve for A:
[pic] Axex + Aex = Axex + ex A = 1. So, the general solution to this differential equation is thus:
y = c1ex + xex.
variation of parameters
1.
[pic] We want to find the general solution to the differential equation, that is, we want to find solutions to the homogeneous differential equation
[pic] From the characteristic equation
[pic] [pic] Since we have a repeated root, we have to introduce a factor of x for one solution to ensure linear independence.
So, we obtain u1 = e−2x, and u2 = xe−2x. The Wronskian of these two functions is
[pic] [pic] Because the Wronskian is non-zero, the two functions are linearly independent, so this is in fact the general solution for the homogeneous differential equation (and not a mere subset of it).
We seek functions A(x) and B(x) so A(x)u1 + B(x)u2 is a general solution of the non-homogeneous equation. We need only calculate the integrals
[pic] that is,
[pic] [pic] where C1 and C2 are constants of integration.
Applications,
The method of undetermined coefficients can be stated in a way which makes it seem more like a unified principle and less like a collection of rules. To do so we must deal with homogeneous equations of higher order than 2. We solve these in exactly the same way we solved second-order equations: write down the charactersistic equation, factor it, and translate the factors into terms of the general solution. Many fundamental laws of physics and chemistry can be formulated as differential equations. In biology and economics differential equations are used to model the behavior of complex systems. The mathematical theory of differential equations first developed, together with the sciences, where the equations had originated and where the results found application. However, diverse problems, sometimes originating in quite distinct scientific fields, may give rise to identical differential equations. Whenever this happens, mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena. As an example, consider propagation of light and sound in the atmosphere, and of waves on the surface of a pond. All of them may be described by the same second-order partial differential equation, the wave equation, which allows us to think of light and sound as forms of waves, much like familiar waves in the water. Conduction of heat, the theory of which was developed by Joseph Fourier, is governed by another second-order partial differential equation, the heat equation. It turned out that many diffusion processes, while seemingly different, are described by the same equation; Black-Scholes equation in finance is for instance, related to the heat equation.
Conclusion,
There is a way of thinking about these rules which is in some sense more satisfying, in terms of annihilation operators. First, some notation: instead of writing y0 , y00, y000 etc. for the derivatives of a function y, we write Dy, D2y, D3y, etc. Then our general nonhomogeneous equation above is written (aD2+bD+c)y = g(x). An an- nihilation operator for a function f(x) is simply some polynomial combination of derivatives anDn+an−1Dn−1+...+a1D+a0 such that (anDn+...+a1D+a0)f(x) =
0. In other words, f(x) is a solution to the homogeneous differential equation defined by anDn + ... + a1D + a0. Then suppose we have differential equation
(aD2 + bD + c)y = g(x), and let P(D) be an annihilation operator for g(x). We get our trial solution for the method of undetermined coefficients as follows: write down the homogeneous differential equation P(D)(aD2 + bD + c)y = 0 associated to the product of P(D) and aD2 + bD + c. We can find the general solution to this equation be exactly the same method as for second-order DEs. We take this general solution, and remove all factors which are annhilated by aD2+bD+c. The result is our guess for the method of undetermined coefficients.
Reference,
1. Differential equations with Maple V- Martha L. Abell, James P. Braselton,
2. A modern introduction to differential equations- Henry Ricardo
