What is Nonhomogeneous: Definition and 80 Discussions

Non-homogeneous Gaussian regression (NGR) is a type of statistical regression analysis used in the atmospheric sciences as a way to convert ensemble forecasts into probabilistic forecasts. Relative to simple linear regression, NGR uses the ensemble spread as an additional predictor, which is used to improve the prediction of uncertainty and allows the predicted uncertainty to vary from case to case. The prediction of uncertainty in NGR is derived from both past forecast errors statistics and the ensemble spread. NGR was originally developed for site-specific medium range temperature forecasting, but has since also been applied to site-specific medium-range wind forecasting and to seasonal forecasts, and has been adapted for precipitation forecasting.
The introduction of NGR was the first demonstration that probabilistic forecasts that take account of the varying ensemble spread could achieve better skill scores than forecasts based on standard Model output statistics approaches applied to the ensemble mean.

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  1. J

    Differential Equation NonHomogeneous EQ

    Homework Statement Find a particular solution of the given differential equation. y'' + 4y = (sin^2)(x) Answer from book: yp(x)= (1/8)(1-xsin2x) The Attempt at a Solution y'' + 4y = 4(sinx)^2....(1) 4(sinx)^2=2 - 2cos2x Homogeneous solution: r^2+4=0 r^2= -4 r=+-2i ==>...
  2. U

    Nonhomogeneous Help on guessing

    hello can someone explain how to guess the yp for this Non-homogeneous differential equation y'' - 2y' + y = te^t characteristic polynomial: (y - 1)^2 so the characteristic roots are: y1=y2= 1 c1 and c2 are constant for yh = (c1)e^t + (c2)te^t please explained how to guess for te^t
  3. J

    Nonhomogeneous Equations; Method of Undetermined Coefficient

    I'm stuck on just one problem. Homework Statement 2y'' + 3y' + y = t2 + 3sin(t) Homework Equations It says in the lesson that if you have a polynomial, guess a solution is "Yi(t)= Ts(A0tn + A1tn-1 + ... + An) where s is the smallest nonnegative integer (s=0,1, or 2) that will...
  4. J

    Initial value nonhomogeneous DE

    Homework Statement (D^3 - D^2 + D - 1)y = e^x and y(0) = 0, y'(0) = 1, y"(0) = 0 Homework Equations D = d/dx The Attempt at a Solution Factoring the left side of the equation gives : (D^2 + 1)(D - 1)y = e^x Which has roots of +/- i and 1. So y(c) = Ae^x + Bsin(x) +...
  5. S

    System of nonhomogeneous difference equation

    How do you solve the system z(t+1)=Az(t)+b where A is a 2x2 matrix and z(t+1), z(t), b are 2x1 matricies? I solved the homogeneous solution: z(t)=P(D^t)(P^-1)z(0) where D is the diagonal matrix of eigenvalues of A and P is the matrix of eigenvectors. I tried to solve the nonhomogeneous...
  6. I

    Nonhomogeneous diff equations method of undeterined coeff.

    Homework Statement Find the general solution to the diff equation using undetermined coefficients y''-2y'-3y = 3te^-1 Homework Equations The Attempt at a Solution r^2 - 2r -3 = 0 r = -1, 3 so y = c1 e^-t + c2e^3t + yp since e^-t already exists as a solution, i have to...
  7. K

    Nonhomogeneous PDE with non-constant coefficients

    This is a question from a book in which I can't figure out, but it has no solutions at the back. Find the general solution to the PDE: xy ux + y2 (uy) - y u = y - x I've learned methods such as change of variables and characteristic curves, but I'm not sure how I can apply them in this...
  8. R

    2nd Order nonhomogeneous ODE using Undetermined Coefficients

    Homework Statement Find General Solution: y"+6y'+9y=e-3x-27x2 Homework Equations The Attempt at a Solution I know you have yh which is the general solution to the left side of the equation set to 0 and then fine the particular solution. When i try to find yp1 I get...
  9. F

    Nonhomogeneous 2nd order dif question?

    y''-2y''-3y=3e^2t find the general solution I have tried Ate^t, Ate^2, Ate^3 none have worked they all leave extra variables that don't match up. is there another combination I could try?
  10. X

    Help solving nonhomogeneous de

    Hey guys just asking for a bit of help to get me on the right track. I have the non homogeneous de 4y" + 4y' + y = 3*x*e^x, which also has some inital conditions y(0)=0 and y'(0)=0. but i only need help with getting the particular solution. Tried method of constant coefficients and it...
  11. S

    ODE ( 2nd order nonhomogeneous equation)

    Homework Statement By using the method of undetermined coefficients,find the particular solution of y''+y'+y=(sin x)^2 Homework Equations i know how to determine the particular solution IF it is sin x. Ex: sin x ====> Asin x + B cos x (particular) but i wonder how to determine the...
  12. H

    NonHomogeneous Equations and Undetermined Coefficients

    NonHomogeneous Equations and Undetermined Coefficients Find the particular solution; y''-10y'+25y=-18e^(5t) here is my work yp(x)=-Ae^(5t) yp'(x)=-5Ae^(5t) yp''(x)=-25Ae^(5t) plug into equation [-25Ae^(5t)]-10[-5Ae^(5t)]+25[-Ae^(5t)... Now; I have 0=-18e^(5t) which doesn't...
  13. Μ

    Fundamental Solution for Nonhomogeneous Heat Equation?

    Homework Statement So I'm trying to solve Evans - PDE 2.5 # 12... "Write down an explicit formula for a solution of u_t - \Delta u + cu=f with (x,t) \in R^n \times (0,\infty) u(x,0)=g(x)" Homework Equations The Attempt at a Solution I figure if I can a fundamental solution...
  14. N

    Nonhomogeneous with constant coefficients equation

    y'' - 3y' + 2y = et + t2 r = 1, 2 -> yc = c1et+c2e2t yp1 = Atet since Aet is a linear combination of our solution to yc. yp2 = At2+Bt+C y'p2 = 2At+B y''p2 = 2A via substitution we have 2A-3(2At+B)+2(At2+Bt+C) = t2 by isolating terms: 2At2 = t2 2A = 1 -> A = 1/2 -6At +...
  15. S

    Nonhomogeneous 2nd order DE

    Homework Statement y'' + 9y = 2x2e3x + 5 Homework Equations N/A The Attempt at a Solution I think the complementary solution yc = c1cos(3x) + c2sin(3x). If not for that little +5 at the end of the right hand side, I'm pretty sure I could solve it. But I don't know how to include it in my...
  16. M

    System of Second Order, Nonhomogeneous Differential Equations

    Hello. I am an engineering student and am having trouble trying to figure out how to solve this system of second order, nonhomogeneous equations. I know how to solve a single second order, nonhomo. equation and how to solve a system of first orders, but not this one. Any help would be greatly...
  17. G

    Nonhomogeneous ODE with Dirac delta

    Trying to solve the ODE mx''(t) + bx'(t) + kx(t) = F(t) with m measured in Kg, b in Kg/s and Kg/s^2, F(t) in Kgm/s^2 and x(t) in m with initial conditions x(0) = 0 and x'(0) = 0, i got the following Green's function G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right]...
  18. C

    Solving Nonhomogeneous Heat Equation with Fourier Transform

    How would one obtain a Fourier Transform solution of a non homogeneous heat equation? I've arrived at a form that has \frac{\partial }{ \partial t }\hat u_c (\omega,t) + (\omega^2 + 1)\hat u_c (\omega,t) = -f(t) My professor gave us the hint to use an integrating factor, but I don't see...
  19. H

    Solving Nonhomogeneous Cauchy Equations with Erwin Kreyszig's WILEY Book

    I got this book from WILEY by Erwin Kreyszig. It tells how to solved homogenous cauchy equations. It also covers simple nonhomogenous equations. But it doesn't cover when we have nonhomogenous Cauchy equations like this one. x2y''-xy'+y=lnx How do I go about solving that equation? I substituted...
  20. S

    Nonhomogeneous Power Series Solution

    For the fun of it, my DE book threw in a couple of problems involving nonhomogenous second order DE's in the section I'm currently going through. Although I have solved for the complementary solution, any suggestions on how to find the particular solution? For example, the one I'm looking at...
  21. z-component

    Linear algebra: nonhomogeneous system

    Homework Statement Let A = \left( \begin{array}{l} \begin{array}{*{20}c} 0 & 1 & { - 1} \\ \end{array} \\ \begin{array}{*{20}c} 2 & 1 & 1 \\ \end{array} \\ \end{array} \right). Suppose that for some b in \mathbb{R}^2, p = \left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ 1 \\...
  22. G

    Nonhomogeneous Boundary Value Problem

    I've got a nonhomogeneous BVP I'm trying to solve. Both my book and my professor tend to focus on the really hard cases and completely skipp over the easier ones like this, so I'm not really sure how to solve it. It's the heat equation in a disk (polar coordinates) with no angle dependence...
  23. S

    Help with nonhomogeneous linear equation

    18.1 #33 Let L be a nonzero real # (a) Show that the boundary-value problem y''+vy=0, y(0)=0, y(L)=0, has only the trivial solution y=0 for the cases v=0 and v<0. I get (a), but I don't know how to do (b) (b) For the cases v>0, find the values of v for which this problem has a...
  24. H

    Nonhomogeneous Linear Differential Equations with Constant Coefficients

    I was wondering if anyone could check my work on this to make sure I'm doing this right for finding a particular solution to y''' + 3y'' + 3y' + y = e^(-x) + 1 + x. First I split the problem into 2 halfs y_p1 and y_p2. y_p1 = Ce^(-x) -Ce^(-x) + 3Ce^(-x) - 3Ce^(-x) + Ce^(-x) = e^(-x)...
  25. Z

    Nonhomogeneous Differential Equation

    I got a particular solution y_p(x) that is different from what the book has. y'' + 9y = 2cos3x + 3sin3x Characteristic equation: r^2 + 9 = 0 (r+3i)(r-3i) = 0 y_c = c_1cos3x + c_2sin3x y_p = Acos3x + Bsin3x (not linearly independent, so I'll try another y_p) y_p = Axcos3x + Bxsin3x...
  26. T

    Analyzing the Linear Nonhomogeneous System

    Let x = x1(t), y = y1(t) and x = x2(t), y = y2(t) be any two solutions of the linear nonhomogeneous system. x' = p_{11}(t)x + p_{12}(t)y + g_1(t) y' = p_{21}(t)x + p_{22}(t)y + g_2(t) Show that x = x1(t) - x2(t), y = y1(t) - y2(t) is a solution of the corresponding homogeneous sytem...
  27. J

    Understanding Arbitrary Constants in Second Order Nonhomogeneous Equations

    The following equation was derived from a RLC circuit: \frac{d^2}{dt^2} (V(t)) + 6 \frac{d}{dt} (V(t)) + 5V(t) = 40 Setting up the equation: s^2 +6s + 5 = 0 yields s = -1 and s = -5 Giving me the general equation: V(t) = k_{1}e^{-t} + k_{2}e^{-5t} But the general equation...
  28. A

    Nonhomogeneous: Undetermined coefficients

    (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...
  29. S

    How Do You Solve a Nonhomogeneous Second-Order Differential Equation?

    Can anyone give me a hand with this, cause I'm stumped and can't remember exactly how to go about solving this. here's the eqn m[d^2x/dt^2 + wsubo^2 x] = F cos wt I'm supposed to show that x(t) = xsubo cos wt w is the incident freq wsubo is the resonant freq m is mass I'm stuck...
  30. W

    Second Order Nonhomogeneous Linear Differential Equations

    Hello, I am having trouble understanding how to solve second order nonhomogeneous linear differential equations. I know how to solve second order homogeneous linear differential equations. But I am not following in the lecture and in the text the method of variation of parameters to solve...
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