What is Coefficients: Definition and 803 Discussions
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression; it is usually a number, but may be any expression (including variables such as a, b and c). When variables appear in the coefficients, they are often called parameters, and must be clearly distinguished from those representing other variables in an expression.
For example,
2
x
2
−
x
+
3
{\displaystyle 2x^{2}-x+3}
, has the real coefficients 2, -1, and 3 respectively, and
a
x
2
+
b
x
+
c
{\displaystyle ax^{2}+bx+c}
, has coefficient parameters a, b, and c respectively assuming x is the variable of the equation.
The constant coefficient is the coefficient not attached to variables in an expression. For example, the constant coefficients of the expressions above are the real coefficient 3 and the parameter represented by c.
Similarly, the coefficient attached to the highest multiplicity of the variable in a polynomial is referred to as the leading coefficient. For example in the expressions above, the leading coefficients are 2 and the parameter represented by a.
The binomial coefficients occur in the expanded form of
(
x
+
y
)
n
{\displaystyle (x+y)^{n}}
, and are tabulated in Pascal's triangle.
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...
Fix some constant 0<\alpha \leq 1, and denote the floor function by x\mapsto [x]. The conjecture is that there exists a constant \beta > 1 such that
\beta^{-n} \sum_{k=0}^{[\alpha\cdot n]} \binom{n}{k} \underset{n\to\infty}{\nrightarrow} 0
Consider this conjecture as a challenge. I don't...
Homework Statement
Find the solution of the given initial value problem.
\[y''-2y'-3y=3te^{2t}\]Homework Equations
The Attempt at a Solution(1) the homogenous solution is given by
\[y_{h}=c_{1}e^{3t}+c_{2}e^{-t}\]
(2) the particular solution is in the form
\[y_{p}=(At+B)e^{2t}\]
(3) The...
for a system defined as y(n) + a1y(n-1) + a2y(n-2) = bx(n), for what values of a1, a2, and b is the system realizable?
i know that the transfer function is obtained by the following
H(z) = Y(z)/X(z)
H(z) = b/(1 + a1z-1 + a2z-2)
H(z) = bz2/(z2 + a1z + a2)
i also know that the poles have...
Homework Statement
An electron with kinetic energy 5 eV (8.01E-19 J) passes through a 3 eV (4.806E-19 J) potential barrier. There are certain widths for this potential barrier in which the transmission probability will equal one hundred percent and the reflection probability will equal zero...
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...
Hi there. I had some trouble trying to solve this:
y''+y=\cos x +3\sin 2x (1)
At first I just found the solution for the homogeneous equation:
y_h=e^{\lambda x} \rightarrow \lambda^2+1=0 \rightarrow \lambda_1,\lambda_2=\pm i
Then y_h=C_1\cos x+ C_2 \sin x
So I've tried to find the particular...
Homework Statement
Hi. I have this problem, which says: The equation x^2y''+pxy'+qy=0 (p and q constants) is called Euler equation. Demonstrate that the change of variable u=\ln (x) transforms the equation to one at constant coefficients.
I haven't done much. I just normalized the equation...
Homework Statement
The function f(x)=ln(10-x) is represented as a power series:
\sum^{\infty}_{n=0}a_{n}x^{n}
Find the first few coefficients in the power series. Hint: First find the power series for the derivative of .
The Attempt at a Solution
Okay, start seems fairly...
Does the variable coeffcient in linear differential equations have to be polynomials? All the examples I have found seem to be polynomials for example the Cauchy–Euler equation.
For example is sin(x)y'' + cos(x^2)y' + y = 0 still considered a linear differential equation with variable...
Homework Statement
Calculate [x^n] (1-2x+x^2)^{(-k)} Homework Equations
Just the geometric series.The Attempt at a Solution
This is what I got so far [x^n] (1-2x+x^2)^{(-k)} = [x^n]((x-1)^2)^{(-k)} = [x^n] \frac{1}{(x-1)^{(2k)}}
Basically, how do I put this into a sum form? or can I just...
Hi, everyone:
If we change the coefficients used to calculate homology, the universal coefficient
theorem tells us how the homology changes. Still, is there a way of knowing whether
a specific (non-trivial) cycle under certain coefficient is still a cycle after the coefficient...
Homework Statement
y-(sinx)2)dx + sinx)dy
Homework Equations
Since the result wouldn't be a line, the equation would only be linear in one of its variables.
The Attempt at a Solution
y-(sinx)2 = 0 ; sinx = 0 --- y = (sinx)2 + sinx
No clue... Also, is there more than one way to...
Homework Statement
The following is the output of a 16 point computation of the discrete Fourier Transform for data taken at 10 kHz. Complete the table below by determining the frequencies associated with each of the FFT values.
I attached an Excel workbook with the table.
Homework...
Hi. I'm trying to make a simple model of a condensation based countercurrent heat exchanger where liquid water and steam flows are separated by a conducting wall. I formed the equations as
Q = \frac{T_{steam}-T_{water}}{\frac{1}{Ah_{water}}\frac{d}{Ak_{wall}}\frac{1}{Ah_{steam}}} = L\dot{m}...
have been solving PDEs by sep of variables, and the solution that comes out is generally a summation the general look of it is something like:
U=SIGMA(n=1 to infinity)E_n(sin(n(pi)x/L)(cos(n)(pi)x/L)t
The above may not be exactly right, I was thinking along the lines of heat equation where...
Hi all!
I'm trying to solve the following system of ODE's, but somewhat unsuccessful...
\dot \vec x = [-i\omega(t)\sigma_z - \nu(t)\sigma_y]\vec x
with sigma_i the Pauli matrices and w(t) and v(t) well-behaved functions of t (actually I also have that w = 1+v). Nevertheless, v(t+T) =...
Homework Statement
Hi...
Don't know if it's actually homework, since it's not, but I hope it's okay to post in here.
I am looking for a paper/website/article of some sort, that might have the derivations of the above mentioned coefficients ?
It's for calculating the figure of merit...
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...
Homework Statement
Prove that a polynomial f of degree n with coefficients in a field F has at most n roots in F.
Homework Equations
The Attempt at a Solution
So we could prove this by induction by using a is a root of f if and only if x-a divides f. My question is: why do...
Hi everyone
I'm modeling the dynamics of a cantilever that has a non-constant linear density profile, i.e.
\rho(x)=\rho_{1} \0 \leq x \leq x_{0}
\rho(x)=\rho_{2} \0 x_{0} \leq x \leq l
\rho(x)=0 \0 otherwise
My differential equation is:
\frac{ d^4 \phi(x) } {d x^4} = \phi(x)...
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...
http://www.cs.odu.edu/~toida/nerzic/content/problem_solving/problem_solving.html
example 3
========================================= Quote
Problem: Given that a, b, and c are odd integers, prove that equation ax2 + bx + c = 0 can not have a rational root.
Understanding the Problem: This is a...
Homework Statement
Find the Clebsch-Gordan coefficients associated with the addition of two angular momenta j_1 = 1 and j_2 = \frac{1}{2}
Homework Equations
The table of coefficients.
The Attempt at a Solution
I think I am misunderstanding something important here. I can't see...
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...
I am trying to find the Fourier coefficients for the following signal:
For some reason, I keep getting 0, which doesn't make sense to me.
I am even getting 0 for F0 even though there is clearly area under the curve. Here's my work for this part:
Period = T/2
Natural freq = 4pi/T
F0 =...
Let V be the space of polynomials of degree 3 or less over \Re. For every \lambda\in\Re the evaluation at \lambda is the map ev_{\lambda} such that V \rightarrow \Re is linear. How do we find the coefficients of ev_{2} in the basis dual to \{1,x,x^2,x^3\}?
Hi all,
I have run a few linear regression models predicting water quality for watersheds using explanatory variables such as mean impervious surface within watersheds and others suggested by theory and the research of others. I would like to be able to compare explanatory variables measured...
The eigenvalue problem of Schroedinger equation can be solved in a variety of ways. The continued fraction method can be stated by the following recipe:
- represent the solution of the D.E. as a power series
- replace back this solution into the D.E.
- obtain a three term recurrence relation...
Right so basically, I'm doing a multicomponent distillation with some exotic chemicals.
My feed comes in at 13kPa and 60*C, so it's under vaccum. This will be a two phase mixture.
What I need is the vapour-liquid-equilibrium coefficients, 'K- values' at feed conditions.
I'm basing this on an...
Homework Statement
Solve the quadratic equation
z^2 + 4(1 + i(3^0.5))z - 16 = 0
Homework Equations
The Attempt at a Solution
I think I've done this correctly, I just wanted to verify.
I've only done the solution for k=0...
Homework Statement
Find the solution of the system
x' = 3x + 4y; y' = -2x-3y
satisfying x(0) = 2 and y(0) = -1
Thank you
Homework Equations
The Attempt at a Solution
My method involves linear algebra:
turn the pair of equations into a single first order vector differential equation of...
Homework Statement
Find the solution of the equation
v''- 4v'+5v=0,such that v=-1 and v'=1 when x=pi=3.14159Homework Equations
...
The Attempt at a Solution
I treat it as a polynomial=>r^2+4r+5=0
=>delta=-4=>r1=2+2i and r2=2-2i
v=e^[x+2](A*cos[2]+B*i*sin[2])
v=-1=e^[pi+2](A*cos[2]+B*i*sin[2])...
Given the difference equation
a_{n+2}+A_n(\lambda)a_{n+1}+B_n(\lambda)a_n=0
where
A_n(\lambda)=-\frac{(n+1)(2\delta+\epsilon+3(n+\gamma))+Q}{s(n+2)(n+1+\gamma)}
and
B_n(\lambda)=\frac{(n+\alpha)(n+\beta)}{2(n+2)(n+1+\gamma)}
The asymptotic behavior of the...
I'm kind of stuck in my search, hoped to be able to find some answers on this board.
Anyways, I've been taksed to design a slow speed mixer and I'm trying to look up information beyond Stokes Law of FD = 6 pi r V u
The fluid to be mixed is approx. 100000 poise, incompressible fluid. I'm...
Homework Statement
Show that if the polynomial p(z)=anzn+an-1zn-1+...+a0 is written in factored form as p(z)=an(z-z1)d1(z-z2)d2...(z-zr)dr, then
(a) n=d1+d2+...+dr
(b) an-1=-an(d1z1+d2z2+...+drzr)
(c) a0=an(-1)rz1d1z2d2...zrdr
Homework Equations
Taylor form of a polynomial? p(z) = \sum...
x y'' + (x + 1) y' = 2 x
Solve for y(x).
Due to the coefficients being a function of x, I have no idea where to start to find the homogenous solution (Complementary Function). I know how to proceed after this part with the variation of parameters method.
I just have no idea where to...
Homework Statement
Find and prove a formula for sum{ (m1 choose r)(m2 choose s)(m3 choose t) }
where the sum is over all nonnegative integers r, s, ant t with fixed sum r + s + t = n.
Homework Equations
The Attempt at a Solution
I first attempted to find the number of combinations of r...
Hi
Few months ago, I had written to PUMA, ADIDAS, NIKE etc. for the shoes with high coefficients
of static friction, so that I can make more informed decision when I go to these shops next time. None of them replied. I know that, these big companies do a lot of research on the shoes. They...
Homework Statement
Prove that for an integer n greater than or equal to 2,
nC1 - 2nC2 + 3nC3 - + ... = 0. (nCm means n choose m)
Also,
2x1 nC2 + 3x2 nC3 + 4x3 nC4 +... = n(n-1)2^(n-2)
Homework Equations
(1+t)^a = 1 + aC1(t) + aC2(t^2) + ...
The Attempt at a Solution
I don't know...
If we have a vector that can be expressed in terms of some finite list of basis elements. If we have an orthonormal basis for a vector space V, then a vector v can be expressed as <v,e1>e1 +...+ <v,en>en. This appears to be widely used for many results (such as Gram-Schmidt), but the...
Homework Statement
Argue that if y=f(x) is a solution to the DE: y'' + p(x) y' + q(x) y = g(x) on the interval (a,b), where p, q, and g are each twice-differentiable, the the fourth derivative of f(x) exists on (a,b).
Homework Equations
The Attempt at a Solution
Its a general...
Homework Statement
For t\in\mathbb{R}, let:
A=\left[\begin{array}{ccc}
-1 & 0 & 0\\
0 & 2 & 2t\\
0 & 0 & 2\end{array}\right]
Get the solution for the general equation: X'=A(t)X
Homework Equations
The Attempt at a Solution
I done many of these problems, all with constant...
Hello!
How do I prove
?
Thank you!
(it can be proven by using the convergence of the Fourier series in L_p-norm, but I want to use the above result to prove the convergence in L_2-norm, so I want to avoid that)
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...
I have a three term recurrence relation
\[
a_0=1,
\]
\[
a_1=p_1(1)a_0,
\]
\begin{equation}\label{recurr}
\begin{array}{ccc}
a_{n}=p_1(n) a_{n-1}+p_2(n) a_{n-2}, && n\ge2.\\
\end{array}
\end{equation}
where
\[...
Homework Statement
You are given that the roots of the auxiliary equation associated with the linear, differential equation
\phi(D)y = 2x- 3xe^{-3x}
are m = \pm2,0,0. Write down the form of a particular solution of the differential equation as predicted by the method of undetermined...
Homework Statement
Hello. I'm taking a course on Mathematical Physics, based on Eugene Butkov's book. I'm having trouble solving a DE with variable coefficients to find Green's Function.
The problem asks to find Green's Function through direct construction.
Homework Equations...