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.
Homework Statement
Find solution for y'' + y' -2y = 2x with initial values of y(0) = 0, y'(0) = 1
Homework Equations
I have found yc = c1*exp(-2x) + c2*exp(x), but finding yp is what I'm having trouble with... AND THEN I'm not so sure how to go about the initial value.
The Attempt at...
Homework Statement
tyʺ+yʹ=4t
Homework Equations
The Attempt at a Solution
The problem that I am having with this problem is I've never been shown how to calculate the particular solution when there is an unknown (t) on the left hand side of the equation. If the problem were...
Greetings,
Regarding the two procedures: undetermined coefficients and variation of parameters, can both procedures be used interchangeably - meaning they both solve (non-homogeneous linear equations)?
Does one method work better in certain situations, if so which method is preferred when...
I am looking for the form of a solution for a second order ODE with right hand side: x (sin x + 2)
I'm thinking the form would be (Ax + B) sin(x) + (Cx + D) cos(x) + Ex + F. Does this seem correct?
Thanks for any help or suggestions!
Hi,
I'm having a bit of trouble with a problem here.
The question is: Use the Method of undetermined coefficients to Find the general solution to th system:
dx/dt = y + e^t
dy/dt = -2x + 3y + 1
I've got the homogenous solution fine, however I'm having a bit of difficulty with the...
Homework Statement
Find the solution for y"+2y'+5y=(e^x)sinx
Homework Equations
The Attempt at a Solution
So far I think I've gotten the solution from the characteristic equation, but I'm having trouble with the particular solution.
For the characteristic equation solution...
Homework Statement
Solve the following initial value problem
y'' - 5y' +6y = x*exp(2x), y(0) = y'(0) = 0
Homework Equations
The Attempt at a Solution
Ive found the complimentary solution to be r = 3, 2,
Yg = C1*exp(3x) + C2*exp(2x) + Yp
But to find Yp is giving me the...
y" + 0.5y' + y = 1-cos(t); y(0) = y'(0) = 0
I used method of undetermined coefficients to get particular solution:
Y(t) = -2sin(t) + 1
To get homogeneous solution, I solved characteristic equation to get complex roots:
r_1,2 = -1/4 +- i*sqrt(15)/4
so homogeneous solution is:
y...
hello everyone, i need some huge help here. here's the equation :
y''+y'-6y=10e^2x-18e^3x-6x-11.
complementary solution:c1e^2x+c2e^-3x
s1={e^2x}
s2={e^3x}
s3=(x,1}
ok since e^2x exists in the complimentary solution, it is therefore a solution, so i multiply it by x to get s'1{xe^2x}...
I know how to solve the following ODE with variation of parameters:
y''+4y=4\sec{\left(2t\right)}.
Is there any way to solve this with undetermined coefficients? So far I have tried Yp=Acos(2t)+Bsin(2t), but that didn't work.
Thanks for the help.
Sup' all?
Ok, I have a quick question (hopefully). I'm trying to use the method of undetermined coefficients, and I keep getting stuck at one specific spot in the method. I'm not exactly sure what I'm doing. Let me try and explain:
The problem is given as:
2y''+3y'+y=t^2+3*\sin t
Which...
Hello,
I have this DE:
y'' + 2y' - 3y = 8ex - 12e3x
when I find homogeneous solution I get
yh = c1ex + c2e-3x;
so now to find the particular solution by method of undetermined coefficients, do I set y to smth like this:
y = y1 + y2
where
y1 = Axex,
y2 = A3x ?
since one...
(d^2x/dt^2)+(w^2)x=Fsin(wt), x(0)=0,x'(0)=0
Hope that's readable. First part is second derivative of x with respect to t. w is a constant and F is a constant. I need to find a solution to this using method of undetermined coeffecients and I'm confused with all the different variables. Anyone...