What is Undetermined coefficients: Definition and 113 Discussions

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. Z

    Conditions for the undetermined coefficients method to fail with sinusoidal input

    If ##p^2-4q<0## then we know that the homogeneous equation has a general solution $$y_g(x)=c_1\sin{kx}+c_2\cos{kx}\tag{3}$$ where $$k=\frac{1}{2}\sqrt{-\Delta}=\frac{\sqrt{4q-p^2}}{2}\tag{4}$$ Suppose we guess at a solution ##y_p## to the non-homogeneous equation...
  2. Davidian

    Variation of parameter VS Undetermined Coefficients

    TL;DR Summary: Variation of parameter VS Undetermined Coefficients Hi all, Suppose we want to solve the following ODE 2y''+y'-y=x+7 with two different methods: undetermined coefficients and variation of parameters. The solutions to the homogeneous problem are given by y_1(x)=exp(-x) and...
  3. T

    Scientific Computing: Method of Undetermined Coefficients in Python

    In a self learning project I am fooling around book https://faculty.washington.edu/rjl/fdmbook/ I want to do some of the computation myself to better understand the concepts but the book is Matlab based and Matlab is too expensive. Does anyone by any chance have some of the codes provided by...
  4. cookiemnstr510510

    Differential Equations-Method of Undetermined Coefficients

    So to answer my question I need to reference another problem, I hope i won't get flagged for this... it is only to make a point about the way I am trying to approach this current problem. the previous problem stated: y''+2y'-y=10 so first I am finding Y_(homogeneous) and going straight to the...
  5. Kaguro

    A differential equation with undetermined coefficients

    Homework Statement Solve the following DE with the method of undetermined coefficients: y'' + 4y = 2cos(3x)cos(x) Homework Equations 2cos(3x)cos(x) = cos(4x) + cos(2x) The Attempt at a Solution Let's split the particular integral into two parts: yp1 and yp2. So yp1 is solution for RHS=cos(4x)...
  6. phantom lancer

    Method of undetermined coefficients -- Help please

    Homework Statement Using the method of undetermined coefficients, determine the general solution of the following second-order, linear, non-homogenous equations. y'' - 4y' + 4y = 2e^(2x+3) Homework Equations I'm not sure what to do from here... Also, I'm new here. How do I use the superscript...
  7. yecko

    Method of undetermined coefficients

    Homework Statement y″ − 2y′ − 3y = t 3 e 5t cos(3t) ---equation 1 Yp = (At3 + Bt2 + Ct + D) e 5t cos(3t) + (Et3 + Ft2 + Gt + H) e 5t sin(3t) ---equation 2 Homework Equations If there is any term in common, then the entire complex of product that is the choice for Y must be multiplied by t...
  8. T

    MHB Non-homogeneous differential equation, undetermined coefficients

    Hi! I've been stuck on this problem for 2 days. I'm new here and I've just spent another hour on trying to figure out Latex, but it always ends up in a mess so I'll try without. I hope that's okay. The equation is y''+6y'+9y=4e-x with the value boundaries (I think that's what it's called in...
  9. C

    Method of Undetermined Coefficients

    Homework Statement Find a particular solution yp of the given equation: y''+y=sinx+xcosx Homework EquationsThe Attempt at a Solution 1) First I got the temporary yp: Asinx+Bcosx+(Cx+D)sinx+(Ex+F)cosx 2) Then I got the associated homogenous eq. check for to check for duplicates: C1cosx+C2sinx...
  10. karush

    MHB -17.2.0 Solve y''+7y'+10y=80 by undetermined coefficients.

    $\tiny{17.2.0}$ $\textrm{ Solve the given equation by the method of undetermined coefficients.}$ \begin{align*}\displaystyle y''+7y'+10y&=80\\ \end{align*} $\textrm{auxilary equation}$ \begin{align*}\displaystyle r^2+7r+10&=0\\ (r+2)(r+5)&=0\\ r&=-2,-5 \end{align*} then is it the...
  11. karush

    MHB -17.2.05 by undetermined coefficients.

    $\textrm{Solve the given equation by the method of undetermined coefficients.}$ \begin{align*}\displaystyle y'' +6y'&=-4xe^{-6x}\\ y_p&=Ax^2e^{-6x}+Bxe^{-6x} \end{align*}ok just wanted get this posted before I leave campus so assume finding zeros is next
  12. karush

    MHB -17.2.9 Solve \ y''-y&=2e^{-x}+3e^{2} by undetermined coefficients

    ok going to see if i can do the steps $\textrm{17.2.9 Solve the given equation by the method of undetermined coefficients.}\\$ \begin{align*}\displaystyle y''-y&=2e^{-x}+3e^{2} \end{align*} $\textit{ textbook answer is:}\\$ \begin{align*}\displaystyle y&=c_1e^{-x}+c_2...
  13. karush

    MHB -17.2.02 - Solve 2nd order ODE using undetermined coefficients.

    $\tiny{17.2.02}$ \nmh{1000} $\textrm{Solve the equation by the method of undetermined coefficients.}\\$ \begin{align*}\displaystyle y''-4y'&=\sin{x}\\ y_p&=A\sin{x}\\ \end{align*} $\textit{ answer}$ \begin{align*}\displaystyle y{\left (x \right )}& = C_{1} + C_{2} e^{4 x} - \frac{1}{17}...
  14. karush

    MHB -17.2.01 Solve the given equation by the method of undetermined coefficients

    $\tiny{17.2.01}\\$ $\textrm{ Solve the given equation by the method of undetermined coefficients.}$ \begin{align*}\displaystyle y''+7y'+10y&=80 \end{align*} $\textit{this is in the form}$ \begin{align*}\displaystyle x^2+7x+10&=0\\ (x+2)(x+5)&=0 \end{align*} $\textit{this is a...
  15. H

    I Undetermined coefficients - nonhomogenous system

    Hello! I am taking a course in differential equations and the book I am using is "Elementary Differential Equations" - E. Boyce & R. DiPrima (tenth edition) My question is about guessing the form of a particular solution to a nonhomogenous system of equations. Given a nonhomogenous system of...
  16. kostoglotov

    Need help understanding an aspect of undetermined coeff's

    imgur link: http://i.imgur.com/8TOXi9t.png I am comfortable with the need to multiply the polynomial in front of e^{2x} by x^3, that makes perfect sense in terms of what the text has already said about how no term in the particular solution should duplicate a term in the complementary solution...
  17. I

    MHB Method of undetermined coefficients

    Use the method of undetermined coefficients to find a general solution of the ODE: $y''+3y'+2y=2x^{2}+4x+5$$r^{2}+3r+2=0$ $r=-2$ and $r=-1$ $y_{h}=c_{1}e^{-2x}+c_{2}e^{-x}$ I'm not sure how to get $y_{p}$ here So here's what I've done so far. I have my final exam tomorrow and I have a few...
  18. kostoglotov

    Trial solution, undetermined coefficients, 2nd order non-homogeneous equation

    Homework Statement Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. For: y'' + 2y' + 10y = x^2e^{-x}\cos{3x} There's a modification performed and I'm not 100% confident as to why. Homework EquationsThe Attempt at a Solution The...
  19. Destroxia

    Undetermined Coefficients, Differential Equations

    Homework Statement y'' + y =3*sin(2t) +t*cos(2t) Okay, so I have found the complimentary solution, and the first partial solution as listed in my work below. My problem is the work on the second partial solution. I have got all the derivatives plugged into the differential equation, my...
  20. S

    MHB Using the Method of Undetermined Coefficients to find a particular solution

    Note that the general solution to $y'' - y = 0$ is $y_h = C_1e^t + C_2e^{-t}$ In the following, use the Method of Undetermined Coefficients to find a particular solution. a)$y'' - y = t^2$ So here is what I have so far $y_p = At^2 + Bt + C$ $(y_p)'' = 2A$ Ive got $A = -1, B = 0 , C = 0$ so...
  21. C

    Method of undetermined coefficients

    Hi all, I have a quick question. I was taught this, but wasn't explained to at all why it is the case. So let's say I have a differential equation with constant coefficients i.e. y'' - 4y' + 4y = e^2x And the general solution to its associated homogeneous equation is Ae^2x + Bxe^2x [A &...
  22. D

    High order differential equations: undetermined coefficients

    Homework Statement If the method of undetermined coefficients is used to find a particular solution yp (t) to the differential equation y'''-y'=te^(-t)+2cos(t) should have the form: ?Homework EquationsThe Attempt at a Solution LHS r^3-r=0 roots= 0, 1 y_c(t)=c_1e^tRHS te^(-t)+2cos(t)...
  23. D

    2nd order differential equation: undetermined coefficients

    Homework Statement y''-y=t-4e^(-t)Homework Equations method of undetermined coefficients The Attempt at a Solution solving for characteristic equation first y''-y=0 r^2-1=0 c_1e^(-t)+c_2e^(t) RHS particular solution t-4e^(-t) y_p(t)= At+B+Ce^(-t) y_pt'(t)=A-Ce^(-t) y_p''(t)=Ce^(-t)...
  24. C

    Systems of Differential Eq (Undetermined Coefficients)

    Hello guys, I need help solving this problem. Find the particular solution using method of undetermined coefficients: X'=AX + F(t) A= [4 ,1/3] <-- 1st row [9 , 6] <-- 2nd row F(t) = [-e^t,e^t] The complementary function is Xc=c1[1,3]e^(3t) + c2[1,9]e^(7t) Any help would be...
  25. K

    Undetermined coefficients

    Hi there, I know that when I am to guess a solution to to a polynomial for g(t) that I guess Ax^n + Bx^n-1... when the highest power of the polynomial is n but what is my guess supposed to be if the power of n is negative? ex. y'' + 4y' + 4y = t^-2*e^(-2t) so far my guess is, A*e^(-2t)(B*?...)
  26. _N3WTON_

    Method of Undetermined Coefficients

    Homework Statement Find the general solution by finding the homogeneous solution and a particular solution. y'' + 4y' = x Homework EquationsThe Attempt at a Solution First, I found the corresponding solution to the homogeneous differential equation: y'' + 4y' = 0 r^{2} + 4r = 0 r_1 = 0...
  27. K

    Using Undetermined Coefficients to solve an equation for a particular solution?

    Homework Statement y'' + 9y = 3sin(3x) + 3 + e^{3x} Homework EquationsThe Attempt at a Solution This is my first post here so let me know if I've done anything wrong, I've been looking at questions here for a long time though ^^. So the problem asks me to solve for one particular solution...
  28. C

    Is It Possible to Use the Method of Undetermined Coefficients Here?

    I'm doing a practice problem I found online, and I get a solution, but I think it should have a sine term in it. I looked up the solution, and most sites say to use variation of parameters, but is it possible to use the method of undetermined coefficients? The problem is as follows: y'' + 4y...
  29. C

    2nd order ODE - undetermined coefficients?

    Homework Statement 1) Find the general solution of y''+ω02=Ccos3(ωx) 2) Show there exists two frequencies at which resonance occurs and determine them The Attempt at a Solution I've tried the method of undetermined coefficients, assuming a solution of the form y=(Acos(ωx)+Bsin(ωx))3...
  30. PsychonautQQ

    Differential Equations Method of Undetermined Coefficients

    Homework Statement consider y'' + 2y' - 3y = 1 + xe^x, find the particular solution The Attempt at a Solution so f(x) = 1 + xe^x f'(x) =e^x + xe^x f''(x) = 2e^x + xe^x so it looks like my particular solution is going to have a constant term, an e^x term and an xe^x term, so I can...
  31. H

    Solving DE by undetermined Coefficients

    Homework Statement y'' + y = xsinx where y(0) = 0 and y'(0) = 1 Homework Equations yc = C1sinx + C2cosx The Attempt at a Solution So my attempt includes the yc which undoubtedly is correct. I now try to solve yp yp = Axsinx + Bxcosx y''p = A(2cosx - xsinx) + B(-2sinx -...
  32. A

    General Question about Undetermined Coefficients

    Hello, I had a question about the method of undetermined coefficients for solving ODE's. I understand it is only useful for certain non-homogeneous functions, and those dictate specific guesses, but what if I had a sum or product of two valid functions, is the guess simply a sum or product of...
  33. MarkFL

    MHB Justifying the Method of Undetermined Coefficients

    The annihilator method can be used to derived the entries in the following table for the method of undetermined coefficients, familiar to all students of ordinary differential equations: Type $g(x)$ $y_p(x)$ (I) $p_n(x)=a_nx^n+\cdots+a_1x+a_0$ $x^sP_n(x)=x^s\left(A_nx^n+\cdots+A_1x+A_0...
  34. C

    Method of Undetermined Coefficients for Higher Order Linear Equations

    Hi, I'd just like to have a quick clarification with regards to the method of undetermined coefficients. I know that if a characteristic equation has the form (r-4)3 = 0 then the characteristic solution will be yc = e4t + te4t + t2e4t + t3e4t and the particular solution ought to be...
  35. 1

    Undetermined coefficients why doesn't it work?

    Homework Statement y''+y=3sin2t+2tcos2t Homework Equations The Attempt at a Solution I tried doing this problem by breaking it into 3 others. It doesn't quite work, and I want to know why. First, I solved for the nonhomog part of: y''+y=3sin2t Which is Y1 = -sin2t...
  36. M

    Undetermined coefficients with annihilator approach (DE)

    There isn't a specific problem that's making me stuck, but I was hoping if someone could point me in the right direction here. I've looked up the topic online, but most of what I could find was through another approach or very unclear. The book I'm using also does not use the method my professor...
  37. A

    Solving a differential equation w/ undetermined coefficients

    Homework Statement y''+2y'+y=xe-x Homework Equations Yc=c1e-x+c2xe-x relevant info on textbook: "If any term of yp is a solution of the complementary equation, multiply yp by x (or by x2 if necessary)." >> i don't understand the part where it says "a solution of the complementary equation"...
  38. D

    MHB Undetermined coefficients

    $$ A\exp\left(\frac{-t}{2}\right)\sin x\cosh \left(\frac{\sqrt{5}}{2}t\right) + B\exp\left(\frac{-t}{2}\right)\sin x\sinh \left(\frac{\sqrt{5}}{2}t\right) $$ What would be the guessed form using the method of undetermined coefficients? Also, if I had $$...
  39. iVenky

    P.I of tanx and secx using Undetermined coefficients

    I just don't get why the method of undetermined coefficients can't be applied to tan(x) and sec(x). What my book says is this- "Since the number of terms applied by differentiating tan(x) and sec(x) is infinity". What do they mean by that? Even the number of terms obtained by...
  40. Hercuflea

    Annihilator method of undetermined coefficients.

    Homework Statement I found everything except step #5. Please tell me if I am correct Find a particular solution to (D - 1)(D^{2} + 4D - 12)y = cos(t) using the annihilator approach of the method of undetermined coefficients. Homework Equations 1) Find annihilator 2) Find A =...
  41. S

    A question on undetermined coefficients

    Homework Statement http://gyazo.com/6c440aa92106f729639c91f6d59dcd89 The Attempt at a Solution My question is why is yp = At^3+Bt^2 +Ct. The reason I ask that is because is see t^2+2t so why wouldn't it be yp=At^2 +Bt +c?
  42. U

    Method of undetermined coefficients, when to raise the guess

    Hello! I have some examples of non-homogeneous ODEs to be solved by the undetermined coefficients method. Two from "Pauls math notes" page: y''+8y'+16y=e^{-4t}+(t^2+5)e^{-4t} The compsol. for this is: Y_{c}=C_{1}e^{-4t}+C_{2}te^{-4t} The first guess for a particular solution would be...
  43. X

    Method of Undetermined Coefficients Question?

    Homework Statement Find a suitable form for the general solution of y'' - 4y' + 4y = 2t2 + 4t*e2t + t*sin(2t) For respective particular solutions, state where it is a general or a special case. DO NOT evaluate coefficents.Homework Equations Y1 = At2 + Bt + C Y2 = (Dt3 + Et2) * e2t Y3 = (Ft +...
  44. C

    Solving y''+3iy'+y=cos(2t) with Undetermined Coefficients

    If I wanted to solve this y''+3iy'+y=cos(2t) using undetermined coefficients. and I make the guess y=Ae^{2it} then i find y' and y'' and then solve for A. I get that A=-1/9 then I take the real part when I multiply it to Eulers formula. But when I plug this back into check it...
  45. C

    Undetermined coefficients

    Homework Statement Show that the imaginary part of the solution of z''+z'+z=te^{it} is a solution of y''+y'+y=tsin{t} The Attempt at a Solution Ok so I first make the guess that z(t)=(at+b)e^{it} then I find z' and z'' and plug it back in and then equate the coefficients of t...
  46. V

    Undetermined Coefficients - Duplication Question

    I'm having some problems recognizing duplication when using the undetermined coefficients method to solve homogeneous type differential equations. http://college.cengage.com/mathematics/larson/calculus_early/3e/shared/chapter15/clc7eap1504.pdf Example 2 on Page 1119, could someone explain...
  47. V

    Solve using Undetermined Coefficients

    y''(t) - \frac{2}{t^2}y(t) = 3 - \frac{1}{t^2} In this problem I had to solve two ways: Variation of Parameters and Undetermined Coefficients. I solved it using Variation of Parameters and got the correct answer for the particular solution in the back of the book being y_p(t) = t^2ln|t| +...
  48. L

    Undetermined Coefficients Problem

    I'm given: y''-y'+\frac{y}{4}=3+e^{\frac{x}{2}} and asked to solve it using undetermined coefficients. Using the auxilary equation y=e^{\lambda t} I got y_{1}=e^{\frac{x}{2}}, y_{1}=xe^{\frac{x}{2}} Now to solve the particular solution, I chose to guess: y_{p}=axe^{\frac{x}{2}}+b and...
  49. R

    Undetermined coefficients

    Homework Statement x(t)^double prime-x(t)=tsint Homework Equations The Attempt at a Solution in this case would I start this as x=t(A(2)+A(1))sint or do i also have to use the cos which makes it x=t(A(2)+A(1))sint+t(B(2)+B(1))cost?
  50. R

    Undetermined coefficients

    Homework Statement determine the particular solution for the differential equation 2x^double prime+x=3t^2 Homework Equations The Attempt at a Solution since F(t)=3t^2 I used At^2+Bt+C and the first derivative is 2A+B plugging back in I get At^2+(4A+B)t+(2B+C)=3t^2 is this...