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.
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...
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...
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...
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...
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)...
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...
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...
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...
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...
$\tiny{17.2.0}$
$\textrm{ Solve the given equation by the method of undetermined coefficients.}$
\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...
$\textrm{Solve the given equation by the method of undetermined coefficients.}$
\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
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...
$\tiny{17.2.01}\\$
$\textrm{ Solve the given equation by the method of undetermined coefficients.}$
\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...
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...
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...
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...
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...
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...
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...
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 &...
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)...
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)...
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...
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*?...)
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...
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...
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...
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...
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...
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 -...
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...
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...
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...
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...
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...
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"...
$$
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)
$$
What would be the guessed form using the method of undetermined coefficients?
Also, if I had
$$...
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...
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 =...
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?
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...
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 +...
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...
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...
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...
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| +...
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...
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?
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...