# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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}...

39. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...