What is Undetermined coefficients: Definition + 113 Threads

In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations 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.
Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.

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  1. O

    Particular Solutions of Undetermined Coefficients

    Homework Statement Find a particular solution y_c to the complex differential equation y'' + 2 y' + 3 y = -1 exp(2 i t) y_c= Using the solution y_c, construct a particular solution y_{1p} to the following differential equation y'' + 2 y' + 3 y = -1 sin(2 t) y_{1p} = Again using...
  2. P

    Method of undetermined coefficients

    solve y''+y'+y=cosx-x2ex y= yc+yp yc: characteristic eq gives r2+r+1=0 using quadratic formula i got r=-1/2+[(sqrt3)/2]i and r=-1/2-[(sqrt3)/2]i so yc=e-x/2(c1cos(sqrt3/2)x+c2sin(sqrt3/2)x but i have no idea of a particular solution that makes any sense. i tried a general...
  3. L

    NonHomogeneous Second Order using Undetermined Coefficients

    Hi all, I understand the basic concept of undetermined coefficients, but am a little lost when g(t) in the equation yll+p(t)yl+q(t)y=g(t) is a product of two functions. The specific problem I'm working on is as follows: yll-2yl-3y=-3te-t When I solve for the homogeneous set of solutions I...
  4. M

    Differential Equations - Method of Undetermined Coefficients

    Homework Statement I've been having problems with a number of these things, here's the first one: y'' -2y' -3y = -3te-t Homework Equations I know that the general solution will be y = yh + yp where yh is the general solution to the homogeneous equation, and yp is the particular...
  5. A

    ODE - having trouble using method of undetermined coefficients

    Homework Statement Find a particular solution. 1. y'' +4y = 4 cos (2t) (for this problem, the instructions tell me that the forcing term is a solution of the associated homogeneous solution) 2. y'' + 16 y = 3 sin (4t) Homework Equations The Attempt at a Solution 1. I guess...
  6. A

    Solving system of differential equations using undetermined coefficients

    Hi, I've come across a problem in my differential equations book that I can't seem to be able to solve (it's not a homework problem, I'm just practicing): "Using matrix algebra techniques and the method of undetermined coefficients, find a general solution for x''(t) + y'(t) - x(t) +...
  7. A

    Method of Undetermined Coefficients help

    Ok, please view the attached image for the Question, and for the given solution. I need some help understanding the solution. I can get the Complimentary equation with no problems, I understand how to do that. However, some questions 1) Why do we first ignore the sin(3x) in our...
  8. B

    Method of Undetermined Coefficients

    [PLAIN]http://www.undeclaredcomics.com/wp-content/uploads/2011/06/rsz_2011-06-06-everyone-gets-it-except-you.png
  9. P

    Method of Undetermined Coefficients

    Homework Statement Decide whether or not the method of indetermined coefficients can be applied to find a particular solution of the given equation: The attempt at a solution I think the answer is yes, because the equation is of the form: a\ddot{y}+b\dot{y}+cy=F(t)where F(t) is the...
  10. C

    Solving y + y = 2/sin(x) using Undetermined Coefficients

    Homework Statement y" + y = 2/sin(x) solve for y Homework Equations I tried to use variation of parameters to solve this but I don't know how to check it. The Attempt at a Solution y = -2xcosx + (constant)cosx + 2ln(sin(x))sinx + (constant)sinx How do I do this using...
  11. B

    What is the Correct Particular Solution for y''-2y'-3y=-3te^-t?

    y''-2y'-3y=-3te^-t i know that that the general solution to this problem is yh = c1e^3t + c2e^-t i am having trouble figuring out what the particular solution is (yp) i keep getting the yp = 3/4te^-t , but wolfram alpha is telling me that the answer is something else. how do i get...
  12. C

    How Do You Solve Nonhomogeneous Equations with Undetermined Coefficients?

    I'm working on undetermined coefficients for nonhomogeneous equations. I have an equation that is equal to 3xe^x. For an earlier one that was equal to e raised to 8x, I used Ae^8x; but on this one, obviously that won't work. I don't know what I should be looking at. I tried Axe^x but no dice...
  13. B

    What Is the Appropriate Particular Solution for y''+25y=2xsin(5x)?

    Homework Statement Set up but do not solve for the appropriate particular solution yp for the differential equation y''+25y=2xsin(5x) using the Method of Undetermined Coefficients (primes indicate derivatives with respect to x). Homework Equations The Attempt at a Solution I first solved the...
  14. J

    How to use of the method of undetermined coefficients

    Correct me if I'm wrong. :) I'm starting to learn basics of the ordinary differential equations, and I have some troubles understanding the concept and method as a whole. I understand when to use the method of undetermined coefficients (MUC) as opposed from variation of parameters. Suppose...
  15. W

    Method of undetermined coefficients problem

    I have the problem "Find the particular solution of 4y''+4y'+y=3xex. What would be the assumed function y? Like would it be Ax ex?
  16. P

    Method of undetermined coefficients

    y''+9y=sin(3t) I need to solve the above using the method of undetermined coefficients. I have already found the solution to y''+9y=0 is c1(Cos3t)+c2(sin3t). The problem is finding the particular solution. From class I am aware that the general form of the solution is...
  17. R

    Undetermined Coefficients, just a piece i with

    Undetermined Coefficients, just a piece i need help with please In one of the last steps in trying to find the Coefficients of Yp (particular solution), I am supposed to set coefficients of similar terms equal to each other. Here's where i need help: i have A(e^x) + (-2B-2C+2F+E)sinx +...
  18. A

    Differential equations - undetermined coefficients

    Homework Statement Find a particular solution of y'' + 2y' + 5y = e^x * sin(x)Homework Equations The Attempt at a Solution Characteristic equation: r^2 + 2r + 5 = 0. Through quadratic formula, r1 = -1+2i and r2 = -1-2i, so my homogeneous equation is y_h = c1 * e^(-x) cos(2x) + c2 * e^(-x) *...
  19. I

    Method of undetermined coefficients question

    Homework Statement y''+y'=g(x) fundamental set of solns. of the homog. DE is: y1= 1 , y2= e^-x If g(x) = x^2 , yp1 = ____ Homework Equations The Attempt at a Solution I actually am looking at a key to an old test my teacher gave as a review for a test. The key has the...
  20. Saladsamurai

    Diff EQ: Method of Undetermined Coefficients

    Homework Statement Solve using Method of Undetermined Coefficients: y'' - 2y' + 3y = 15t - 8 The Attempt at a Solution First I used the characteristic equation to solve for the homogeneous solution to get: yh = C1e-t + C2e3t Great. Now I need to find the particular. I am...
  21. D

    D.E.s Undetermined Coefficients

    Homework Statement y''' + y'' - 6y' = 36x Homework Equations y'' + y' - 6y =18x^2 The Attempt at a Solution I've integrated the third order D.E. to get a second order D.E. Solving the characteristic equation, to find a general solution, I get: y = c_1exp(-3x) + c_2exp(2x)...
  22. D

    Differential Equations, Undetermined Coefficients

    Homework Statement y'' + 3y' + 2y = 24exp(-2x) Homework Equations y'' + 3y' + 2y = 24exp(-2x) The Attempt at a Solution I've solved the characteristic equation and am after the particular solution. I've let y_p = Aexp(-2x), y_p' = -2Axexp(-2x) and y_p'' = 4Axexp(-2x) When...
  23. G

    ODE: Combining Undetermined Coeff. & VOP Method

    Title should read "Combining", is there anyway a moderator could alter that so the search function isn't messed up? Homework Statement The Attempt at a Solution I am familiar with both methods, however combining the two is foreign to me. Anyone have any suggestions for this ODE? My...
  24. J

    Solving x'=Ax+g using undetermined coefficients

    Homework Statement Problem #4 from here: http://www.math.washington.edu/~jtittelf/ExtraCredit.pdf Homework Equations ? The Attempt at a Solution There's only one example of using undetermined coefficients in my textbook. I'm not sure where to start. x = t2a + tb + c ? That's...
  25. M

    Undetermined coefficients - deriving formula

    Homework Statement Derive a formula of the form \int\stackrel{b}{a}f(x)dx \approx c_{0}f(a)+c_{1}f(b)+c_{2}f'(a)+c_{3}f'(b) that is exact for polynomials of the highest degree possible. Apply a change of variable: y=x-a Homework Equations The Attempt at a Solution I don't get the "highest...
  26. K

    Method of undetermined coefficients

    Homework Statement Given are the points (1,-4), (3,0), and (4,1). I am to usemethod of undetermined coefficients. Homework Equations Given these points I am to find a quadratic equation. The Attempt at a Solution Sorry! But I really do not have any clue. May anyone help me?
  27. D

    Undetermined Coefficients problem

    Homework Statement Find the unique solution to the differential equation y''+4y'−5y=−435sin(2t) satisfying the initial conditions y(0)=29 and y'(0)=47 The Attempt at a Solution So (I think) I found the correct general solution. I let y(t)=Acos(2t)+Bsin(2t) and eventually worked...
  28. L

    Undetermined Coefficients, more than one term on RHS

    Homework Statement y''-49y=7cos(7x)+7+e^(7x) The Attempt at a Solution I have no idea how to solve this Differential equation. I could solve one that has y''-49y=one term, but I'm stumped with more than one. First, I get the homogeneous equation, y''-49y=0 and fine y_c, then use the...
  29. K

    Undetermined Coefficients of Difference Equations

    I am working on a section on undetermined coefficients of a difference. I'm really trying to understand the material, but am struggling. y_{n+1}-2y_{n}=2 I got as far as saying for the homogenous equation y=c_{1}(-1)^{n}. I'm assuming the guess next would be A, but I'm confused. Maybe if...
  30. S

    Undetermined Coefficients Initial Value Question

    Homework Statement y''-4y=60e4t y(0)=9 and y'(0)=2 Homework Equations none in particular The Attempt at a Solution First I solved for the auxiliary equation. r2-4r r(r-4) r=0, 4 So the general solution for the homogenous form is C1e0x+C2e4x where C1 and C2 are unknown coefficients to...
  31. D

    Method of undetermined coefficients

    Homework Statement http://edugen.wiley.com/edugen/courses/crs3098/rc/boyce3346c07/math/math1074.gif solve using the method of undetermined coefficients Homework Equations v(t)=a*t*e^-t + be^-t + ct + d The Attempt at a Solution what i could do is that i replaced v(t) for x in the...
  32. O

    Inhomogeneous diff EQ, undetermined coefficients

    Find the solution of: y" + 3y' = 72sin(3t) + 36cos(3t) where y(0) = 6 and y'(0) = 9 I first found the solution to the homogeneous eq: the roots (R^2 + 3R = 0) are R = 0, -3 which gives the family of solutions: y = a(1) + be^(-3t) and y' = -3be^(-3t) using the initial conditions...
  33. L

    Help finding the answer to DE using method of undetermined coefficients

    Homework Statement y'' - 5y' = 2x^3 - 4x^2 -x + 6 The Attempt at a Solution I first found the m's on the left hand side. They are 0 and 5. I then found the m's on the right hand side. They are 0, 0, 0, and 0. I then wrote my interim equation: y = C1 + C2e^5x + Ax^4 + Bx^3 +...
  34. B

    Method of undetermined coefficients

    Homework Statement y'' + y = sin x + x*cos x Homework Equations The Attempt at a Solution i don't like undetermined coefficients. so far, for me, it's been so much trial and error, although i am getting a little faster at it. anyway, i didn't work out the homogeneous equation...
  35. D

    Method of undetermined coefficients Problems

    been working on these all week some help please.. 9) 3y''-6y'+30y= 15sinx+exp(x)tan(3x) 10) y''-2y'+y= 4x^2 -3+ Exp(x)/x[b]2. Homework Equations Variations of parame Method of undetermined coefficients thoughts: i think that you need to split the eqtuaion into 2 parts. Using variation of...
  36. J

    Series solution for ode by undetermined coefficients

    Homework Statement Obtain the Taylor series solution up to and including order 3 of the following non linear ode y'=x^2+\sin y,y(0)=\frac{\pi}{2} Homework Equations After substituting the power series form of sin(y) I get: y'=x^2+(y-\frac{y^3}{3!}+\frac{y^5}{5!}-\frac{y^7}{7!}...)...
  37. P

    Method of Undetermined Coefficients

    Homework Statement y(4)-4y2=t2+et Determine the general solution Homework Equations The Attempt at a Solution So I worked it all out and got the correct answer. My question is that when we make the onsat that a particular solution will be in the form Y=At4+...how do we know it...
  38. M

    O.D.E. Undetermined coefficients solution

    Homework Statement i have this problem y'' - y' - 2y= -6xe^{-x} Homework Equations this is the homogenous solution : y_{h}= c_{1} e^{-x} + c_{2} e^{2x} The Attempt at a Solution first i can do the problem but what is perplexing me is why of these two tentative...
  39. V

    Use of method of undetermined coefficients

    Use of "method of undetermined coefficients" Hi all, Suppose I have a equation f(z+1) - f(z) = z^{1/2} , \forall z \geq 0 eq (1) then is it possible to solve this equation by the method of undetermined coefficients ? It is usually seen in textbooks that the forcing function is taken to be...
  40. R

    2nd Order nonhomogeneous ODE using Undetermined Coefficients

    Homework Statement Find General Solution: y"+6y'+9y=e-3x-27x2 Homework Equations The Attempt at a Solution I know you have yh which is the general solution to the left side of the equation set to 0 and then fine the particular solution. When i try to find yp1 I get...
  41. S

    Where Did I Go Wrong With Undetermined Coefficients for y+y'+4y = 2sinh(T)?

    i solved homogenous but y"+y'+4y = 2sinh(T) is giving me trouble also have a hint sinh t = (e^t - e^-t)/2 also my "guess" was aE^t +Be^-t=Y maybe my guess was wrong?
  42. M

    Solving Systems of DEs with Undetermined Coefficients

    Homework Statement So I'm doing a systems of DEs question with undetermined coefficients. After comparing coefficients, I have this: a+b=Ab+(1,0)T where a&b are vectors (2D) and A is the coefficient matrix: [2 -1] [3 -2] I know that 'a' is (1,1)T. I need to find b. I tried...
  43. K

    Undetermined Coefficients to solve

    Undetermined Coefficients to solve. y'''-6y''=3-cosx So I set everything up, get my system & I get an answer of y=c1+c2x+c3e^(6x)-6/37cosx+1/37sinx the book has y=c1+c2x+c3e^(6x)-6/37cosx+1/37sinx-1/4x^2. So I'm not sure where the book gets the last term, any ideas?
  44. H

    NonHomogeneous Equations and Undetermined Coefficients

    NonHomogeneous Equations and Undetermined Coefficients Find the particular solution; y''-10y'+25y=-18e^(5t) here is my work yp(x)=-Ae^(5t) yp'(x)=-5Ae^(5t) yp''(x)=-25Ae^(5t) plug into equation [-25Ae^(5t)]-10[-5Ae^(5t)]+25[-Ae^(5t)... Now; I have 0=-18e^(5t) which doesn't...
  45. D

    Solution to nonhomogenous DE using Method of Undetermined Coefficients

    Homework Statement Find the general solution of the given differential equation: y'' + 9y = (t^2)(e^3t) + 6 Homework Equations The Attempt at a Solution I want to first find a particular solution using the method of undetermined coefficients, but I'm not sure what I should...
  46. S

    How do you solve a differential equation using undetermined coefficients?

    How do I go about solving a DE using the method of undetermined coeficients in a question like for example; y''-4y'+4y=2e^2x I tried assuming the yp to be Ae^2x but when I plugged it into the DE I ended up with 0=2e^2x ??
  47. S

    Method of undetermined coefficients

    Hi, When using the method of undetermined coefficients to solve 2y'' + y' = cos(2x) + 4x + 1 I did it term by term. I figured out the first yp1 for the cos(2x). I'm just not sure what to "guess" as to the form of the yp2 for 4x+1 Does anyone know what to use as the guess?
  48. N

    Method of Undetermined Coefficients - Inhomogeneous DE

    Given y'(t) + i*k*y(t) - i*g*x(t) = 0 What "form" does x(t) take?
  49. B

    Undetermined coefficients help

    Hi, new here, I am looking for some help on this homework problem. pretty sure you need to separate yp into 2 parts? anyways, use method of undetermined coeff. to solve y" + y = x*cos(x) - cos(x) thanks.
  50. H

    (Methods)Parameters v Undetermined Coefficients

    [SOLVED] (Methods)Parameters v Undetermined Coefficients Can anyone tell me why I would use one technique over the other? It seems as though undetermined Coef. is much easier to do but I suppose that comes with limitations?
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