# 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. ### I Question about solving linear first order non-homogeneous ODEs

A general equation for linear first order non-homogeneous ODE is: ## y' + a(x)y = b(x) ##. The procedure to solve ( assuming ## a(x) , b(x) ## are continuous so that the fundamental theorem of calculus could be used ) it is to multiply it by ## e^{A(x)} ## ( here ## A'(x) = a(x) ## ) s.t. ##...
2. ### I "Rationale" for Homogeneous vs. Nonhomogeneous Differential Equations?

Hi; I am missing something. I can follow the technicality of a homogenous linear equation has all coefficients of zero and the "contra" for non homogenous equations. I just can't figure out the relevance of the consequences of outcome. If I am not being clear maybe I can be guided as to how...
3. ### Finding the general solution of this nonhomogeneous linear system

Hi, I was trying to do the following problem. My attempt. Finding the reduced row echelon form for the system above. I do not see any way to proceed any further. The following is the solution presented in solution manual. How do I proceed to get the following answer?
4. ### A Solution of nonhomogeneous heat equation problem

This is from Evans page 50. I'm sure it's something simple, but I don't follow the change from $$\frac{\partial}{\partial t} \quad \text{to} \quad -\frac{\partial}{\partial s}$$ and from $$\Delta_x \quad \text{to} \quad \Delta_y$$. \begin{gather*} \begin{split} u_t(x,t) - \Delta u(x,t) & =...
5. ### 2nd-order Nonhomogeneous Differential Equation

Homework Statement Finding the general solution: y”+4y’+4y=t*e^(-2t) Homework EquationsThe Attempt at a Solution So I got the complementary solution pretty easily as y= c1*e^(-2t)+c2*te^(-2t) I haven’t been able to find a particular solution using the method of undetermined coefficients. I...
6. ### 2nd-order nonhomogeneous equation

Homework Statement Homework Equations 2nd order nonhomogeneous equation The Attempt at a Solution (answers typed in in the pic) for part c, if I can't put Ate^(2t) & Be^(2t), how is the form of particular equation like? I have used Y=(At+B)e^(2t)+(Ct+D), because of the largest power of the...
7. ### I Separation of variables for nonhomogeneous differential equation

Hi. I was wondering if it is possible to apply separation of variables for a function of space and time obeying a non homogeneous differential equation. In particular, the heat equation: ##\displaystyle \frac{\partial \Phi(\mathbf{r},t)}{\partial t}-\nabla \cdot \left [ \kappa(\mathbf{r})...
8. ### Laplace transform to solve a nonhomogeneous equation

Mod note: Moved from a Homework section can i use the Laplace transform to solve a nonhomogeneous equation if i have these Initial condition s(x) and s(-x)
9. ### A Nonhomogeneous second order nonlinear differential equations

Hello every one, I have an equation related to my research. I wonder if anyone has any suggestion about solving it? y''+y' f(y)+g(y)=h(x) thanks
10. ### Solve a system of nonhomogeneous DEs

Homework Statement Solve $$\ddot {\vec a}=A\vec a+B\dot{ \vec a}+\vec F$$ if A and B are known matrices and F is a constant vector. Homework EquationsThe Attempt at a Solution My plan was to define ##\vec b=\dot{\vec a}## to move from second order to first order system ##\dot{\vec b}=A\vec...
11. ### Dynamic Damping in a simple spring system

Homework Statement imgur link: http://i.imgur.com/Bv3qtPm.png Homework EquationsThe Attempt at a Solution From the FBD it is apparent that there is a constraint -k_1x_1 + k_2(x_2-x_1) + 5\cos{10t} = 0 If you combine this with {x}''_1 = -(k_1+k_2)x_1 + k_2x_2 + 5\cos{10t} and m{x}''_2...
12. ### Nonhomogeneous BC Heat Conduction

Homework Statement A very long cylinder has temperature T1 impressed on half of its peripheral surface and temperature T2 impressed on the other half. Find T(r, θ).Homework Equations governing eq 1/r ∂/∂r(r∂/∂r)+∂2T/∂z2 The Attempt at a Solution I am right at r=0 ;T=T1 at r=r' T=T2 to make BC...
13. ### Insights Solving Nonhomogeneous Linear ODEs using Annihilators - Comments

Mark44 submitted a new PF Insights post Solving Nonhomogeneous Linear ODEs using Annihilators Continue reading the Original PF Insights Post.
14. ### Particular solution to linear nonhomogeneous equation

Homework Statement I started learning about solving non homogeneous linear differential equations in class and I am a bit clueless on how to solve them since I've never had a prior experience with much of differential equation. I am trying to find the particular solutions to the equation...
15. ### Derive gen sol of non-homogeneous DEs through linear algebra

Hello, I noticed that the solution of a homogeneous linear second order DE can be interpreted as the kernel of a linear transformation. It can also be easily shown that the general solution, Ygeneral, of a nonhomogenous DE is given by: Ygeneral = Yhomogeneous + Yparticular My question: Is it...
16. ### Nonhomogeneous System: Similar Coefficients & Solutions?

This might belong in the HW section, but since it's specific to Linear Algebra I posted it here. Alright, so we have a homogeneous system of 8 equations in 10 variables (an 8 x 10 matrix, let's call it A). We have found two solutions that are not multiples of each other (lets call them a and...
17. ### Nonhomogeneous 2nd order dif equation

Hello guys . I want to do nonhomogeneous 2nd order dif equation..I am trying to do this for 2 days , but I can't get good answer. Can you show me how to do this equation with constant variation method ( i know it best) or other I would be very gratefull , because after 2 days of trying I am...
18. ### Nonhomogeneous recurrence relations

Homework Statement Solve the recurrence relation an = 3an−1 −2an−2 +3, a0 = a1 = 1.Homework Equations an = general solution + particular solutionThe Attempt at a Solution I started with finding the general solution, which was easy. it ended up being A12n + A0 now I am having trouble solving...
19. ### Differential Equation, nonhomogeneous equation

Homework Statement Find the general solution: y'' - y' - 2y = -2t + 4t^2 Homework Equations The Attempt at a Solution r_1 = 2, r_2 = (-1) Set Y(t) = At^2 + B^t + C Y' = 2At + B Y'' = 2A 2A - 2At + B - 2At^2 + Bt + C = -2t + 4t^2 -2At^2 + (B-2A)t + 2A + B + C = -2t...
20. ### Second order nonhomogeneous ODE

Homework Statement y''+3y'+3.25=3cost-1.5sintHomework Equations yh = e(a/2)t(Acost+Bsint) yp = Kcos(ωt)+Msin(ωt) [when r(x)=kcos(ωt) or ksin(ωt)]The Attempt at a Solution I got the homogeneous solution, which is e-1.5t(Acost+Bsint) but I am having trouble with the particular solution. I...
21. ### 1st order, nonhomogeneous, linear DE - particular solution

I know nothing about DEs, so this may be a silly question. I'm given some time varying (x_t)_t and a constant r, and I want to solve the equation u_t = rx_t + \dot u_t for u. What I know so far is that (solving the homogeneous equation) if \bar u is some particular solution, then any u is...
22. ### Solving a nonhomogeneous 2nd order ode

Hi, everyone! This is my first post here, I need an hand with this equation! Homework Statement Solve the initial value problem: $$\begin{cases} u''(x)+4u(x)=\cos(2x) \\u(0)=u'(0)=1 \end{cases}$$ The Attempt at a Solution I started by solving the...
23. ### Nonhomogeneous system particular solution.

Homework Statement Verify that the vector functions x_{1}=\begin{bmatrix}e^{t}\\ e^{t}\end{bmatrix} and x_{2}=\begin{bmatrix}e^{-t}\\ 3e^{-t}\end{bmatrix} are solutions to the homogeneous system x'=Ax=\begin{bmatrix}2 & -1 \\ 3 & -2 \end{bmatrix} on (-\infty ,\infty ) and that x_{p}...
24. ### [PDE] Transforming Nonhomogeneous BCs into Homogeneous Ones

So there's this problem in my text that's pretty challenging. I can't seem to work out the answer that is given in the back of the book, and then I found a solution manual online that contains yet another solution. The problem is a the heat equation as follows: PDE: u_{t} = α^2u_{xx} BCs...
25. ### Nonhomogeneous system of linear equations

Hi all, How do u go about doing this question? x - 2y +z =4 y- z =3 (a^2 - a - 2)z = a+1 Determine values of a for which the system has no solution, one solution and many solutions Stephen
26. ### Nonhomogeneous difference equation question

Homework Statement Find the general solution Homework Equations x(t+2)-3x(t+1)+2x(t)=3*5^t+sin(0.5πt) The Attempt at a Solution I start out by solving the homogeneous equation and end up with the two roots 1 and 2. Then I try to use the method of undetermined coefficients to find a...
27. ### Prove that a closed interval [0,1] is nonhomogeneous

Homework Statement Prove that the closed interval [0,1] is not a homogeneous topology by showing that there's no bijective, open and continuous (bi-continuous) mapping h: [0,1]→[0,1] such that h(1/2)=0.Homework Equations The closed interval is equipped with the usually metric. If the mapping...
28. ### Nonhomogeneous Power Series Solution

Hi. I have to solve: y''+xy'-2y=e^x Using series. So, this is what I did: y(x)=\sum_0^{\infty}a_n x^n y'(x)=\sum_1^{\infty}n a_n x^{n-1} y''(x)=\sum_2^{\infty}n(n-1) a_n x^{n-1} And e^x=\sum_0^{\infty}\frac{x^n}{n!} Then, using that m=n-2 for y'' and then replacing in the diff. eq...
29. ### Diff Eq- Nonhomogeneous Equations

Homework Statement Find a particular solution of the given equation. y^''' + 4y^' = 3x-1 Homework Equations r^3 + 4r = 0 r = 0, r = 2i, r = -2i The Attempt at a Solution y(x) = Ax-B y^'(x) = A y^''(x) = 0 y^'''(x) = 0 The answer is: y(x)=(3/8)x^2 - (1/4)x But I'm not...
30. ### Nonhomogeneous linear differential equation

Homework Statement solve for y(x). y"'-6y"+11y'-6=e^{4x} Homework Equations Wronskian determinant. Method of variations. The Attempt at a SolutionSupposing that [u', v', w'] are the solutions, wronskian det=W is 10e^{6x} By use of x_k=\frac{det(M_{k})}{det(x)}, I got...
31. ### Nonlinear nonhomogeneous ODE of the first kind

Homework Statement Solve the following ODE: du/dx=u^2+1 Homework Equations The Attempt at a Solution I have tried making the substitution: u^2=v but this doesn't help. Any hints will be very much appreciated
32. ### Second Order Differential Nonhomogeneous Equation

Homework Statement Use the method of undetermined coefficients to find one solution of http://img85.imageshack.us/img85/6844/4ab921ad6ba6851cc91401c.png Note that the method finds a specific solution, not the general one. Homework Equations Y = Yc + Yp Yc = C1e^(r1t)+C2e^(rt) when...
33. ### Second order nonlinear nonhomogeneous differential equation

Hello, I am having a little trouble solving this equation: \frac{d^2y}{dx^2} + \frac{A}{y}(\frac{dy}{dx})^2 + \frac{B}{(y+C)^2} = D - Ex where A, B, C, D, and E are constants and, sadly, not related. So far, I've found this http://eqworld.ipmnet.ru/en/solutions/ode/ode0344.pdf...
34. ### NonHomogeneous Second Order using Undetermined Coefficients

Hi all, I understand the basic concept of undetermined coefficients, but am a little lost when g(t) in the equation yll+p(t)yl+q(t)y=g(t) is a product of two functions. The specific problem I'm working on is as follows: yll-2yl-3y=-3te-t When I solve for the homogeneous set of solutions I...
35. ### Solution of a nonhomogeneous equation

(i d0nt kn0w how to use LaTeX) 1)D^2(D-1)y=3e^x+sinx for yc: let y=e^mx D^2(D-1)e^mx=0 m^2(m-1)e^mx=0 f(m)=0 m^2(m-1)=0 m=0,0,1 yc= C1+C2x+C3e^x for yp: R(x)=3e^x+sinx m'=1,+/- 1i yp=Axe^x+Bcosx+Csinx yp'=A(xe^x+e^x)-Bsinx+Ccosx yp"=A(xe^x+2e^x)-Bcosx-Csinx D^2(D-1)yp=3e^x+sinx Guys can...
36. ### Nonhomogeneous ODEs that can't be made homogeneous?

Assuming knowledge of homogeneous ODEs and nonhomogeneous ODEs that can be made homogeneous (eg, y'-y=x), how does one solve those that cannot be made homogeneous (eg, y'-y=cosx, y''-xy'+y=0, cos(y'')+sin(y')=0)? EDIT: Maybe "made homogeneous" is the wrong way to put it... By being able to be...
37. ### First oreder linear ODE NONHOMOGENEOUS

Homework Statement x^3y'+xy=x, y(1)=2 Homework Equations The Attempt at a Solution I'm having trouble starting this because it doesn't fit any form I'm familiar with because of the x^3 in front of the y'. Can someone give me some pointers to get started..
38. ### 1st order nonhomogeneous dif'l e'n

f'+x*f=x^(2m+1), m is an integer I have no idea for finding particular solution. Any help would be appreciated.
39. ### Heat equation with nonhomogeneous boundary conditions

Homework Statement Consider \frac{\partial u}{\partial t} = k\frac{\partial^2u}{\partial x^2} subject to u(0,t) = A(t),\ u(L,t) = 0,\ u(x,0) = g(x). Assume that u(x,t) has a Fourier sine series. Determine a differential equation for the Fourier coefficients (assume appropriate continuity)...
40. ### Second-Order Nonhomogeneous DE

The Equation: x^2 (d^2y/dx^2) + 3x (dy/dx) - 3y = x The boundary conditions: y(x=1) = 0 y(x=2) = 1 It's been awhile since I took diffeq, but my research has led me to believe that this is not a Cauchy-Euler Equation since it is not equal to 0, it cannot be separated for separation...
41. ### Second-order nonhomogeneous diff-eq

Hey guys. It's been a few years since I've taken diff-eq and I can't remember how to solve second-order problems like this one: Find the general solution of u'' + a^2*u = cos(bx) I know that if it were homogeneous, I would solve for r^2 + a^2 = 0, and get u = ce^(rx). But for the life of...
42. ### Linear, nonlinear, homogeneous, nonhomogeneous PDE's

1. For each of the following equations, state the order and whether it is nonlinear, linear inhomogeneous, or linear homogeneous; provide reasons. (a) ut-uxx+1=0 (b) ut-uxx+xu=0 (c) ut-uxxt+uux=0 (d) utt-uxx+x2=0 (e) iut-uxx+u/x=0 (f) ux(1+u2x)-1/2+uy(1+u2y)-1/2=0 (g) ux+eyuy=0 (h)...
43. ### Nonhomogeneous second order linear differential equations

Homework Statement i'm supposed to find the general solution of the equation: y'' + 3y' + 2y = e^x + e^-x Homework Equations I have no problem with solving this equation however, i am confused with the step they are taking in the solutions (circled)...
44. ### Variation of Parameters Nonhomogeneous Differential Equation

Homework Statement 4y'' + y = cosx Solve using variation of parameters Homework Equations The Attempt at a Solution from a) -> yc(x) = c1cos(x/2) + c2sin(x/2) let y1 = cos(x/2) , y2 = sin(x/2) y1y2' - y2y1' = 1/2cosx/2 + 1/2sinx/2 = 1/2 u1' = ? How do I find this?
45. ### Second Order Linear Nonhomogeneous Differential Equations

Homework Statement y''+7y'=392sin(7t)+686cos(7t) with y(0)=4 and y'(0)=9 Homework Equations No real relevant equations The Attempt at a Solution I assumed since the g(t) has function of both sine and cosine the solution would be both the real and non real parts of the solution to...
46. ### Differential equations - nonhomogeneous

Homework Statement Let Ly = y'' + py' + qy. Suppose y1 and y2 are functions such that Ly1 = f(x) and Ly2 = g(x). Show that the sum y = y1 + y2 satisfies the nonhomogeneous equation Ly = f(x) + g(x). Homework Equations Superposition Principle: L[c1y1 + c2y2] = c1L[y1] + c2L[y2] The...
47. ### Second order linear differential equations nonhomogeneous equations

I could not get LaTex to format properly, so I typed out the question and my work using Microsoft Word's equation editor. Please see the attached document, apologies for any inconvenience! These problems involve the techniques for the method of undetermined coefficients and variation of parameters.
48. ### Second order differential nonhomogeneous equation with gaussian term

Hello forum, I am trying to solve a differential equation for the last four hours and I can't figure out how... here it is \frac{d^2x(t)}{dt^2} + \frac{dx(t)}{dt} + c x(t) = d e^{-a t^2} actually my problem is how to handle the Gaussian term... if anyone can help please...
49. ### Nonhomogeneous Second Order ODE containing log

Hi guy, I have this ODE that I'm having problems with y"+4y'+4y= e^(-2x)logx Now, Using method of UC to get rid of the RHS I've tried using Ae^(-2x) x^2 logx However, I'm not quite sure whether that is correct or not as I have never had a question containing logs before
50. ### Nonhomogeneous wave equation with vanishing initial conditions

Homework Statement Let u(x,t) be the solution of the following initial value problem for the nonhomogeneous wave equation, u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=f(x_1,x_2,x_3,t) u(x,0)=0 and u_t(x,0)=0 x\in\Re^3 , t>0 Use Duhamel's Principle and Kirchoff's formula to show that...