# linear

1. ### I Find the practical resonance frequencies in a system of linear differential equations

Hi all, I would like to know what is the equation upon which I can use to determine the practical resonance frequencies in a system of second order, linear differential equations. First some definitions: What I mean by practical resonance frequencies, is the frequencies that a second order...
2. ### I Area divided by linear function

The original problem for anyone that can read Chinese: https://zerojudge.tw/ShowProblem?problemid=b221 The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...
3. ### I Linearity of DE's

Hey all, I don't understand what makes a differential equation (DE) linear. I found this: "x y' = 1 is non-linear because y' is not multiplied by a constant" but then also this: "x' + (t^2)x = 0 is linear in x". t^2 also isn't a constant. So why is this equation linear?
4. ### Find the distance from the point P to a line - linear algebra

1. Homework Statement Find the distance from point P (1,7,3) to the line (x,y,z) = (-2,1,4) + s(1,-3,4), s is a free variable 2. Homework Equations projnQP = ( QP⋅n/(lengthQP)(lengthn) )(n) 3. The Attempt at a Solution I'm not quite sure about how to find the normal (n) here, but if I make...
5. ### Angle between lines, with free variables in equations?

1. Homework Statement Find the cosine of the angle determined by the intersecting lines m1: (x,y,z) = (-2,1,4)+s(1,-3,4) m2: (x,y,z) = (-2,1,4)+t(-2,3,2) s and t are free variables 2. Homework Equations a⋅b = length(a)length(b) cosθ 3. The Attempt at a Solution I just did this equation...
6. ### Show that eigenvalue of A + eigvalueof B ≠ eigvalue of A+B?

1. Homework Statement Let A and B be nxn matrices with Eigen values λ and μ, respectively. a) Give an example to show that λ+μ doesn't have to be an Eigen value of A+B b) Give an example to show that λμ doesn't have to be an Eigen value of AB 2. Homework Equations det(λI - A)=0 3. The...
7. ### I Linear fitting in physics experiments with errors

Hello! I have a question that maybe has to do more with Mathematics, but if you do experimental physics you find it quite often. Let's assume that we want to measure two quantities x and y that we know that they relate to each other linearly. So we have a set of data points xi and yi...
8. ### A Nonlinear operators vs linear

Hi, I have in a previous thread discussed the case where: $$TT' = T'T$$ and someone, said that this was a case of non-linear operators. Evidently, they commute, so their commutator is zero and therefore they can be measured at the same time. What makes them however...
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### B Linear combinations

Hi, I read that linear combinations of a state, Psi, can be as: $$\Psi = \alpha \psi + \beta \psi$$ where $\alpha$ and $\beta$ are arbitrary constants. Can however this be a valid linear combination? \Psi = \alpha \psi \times \beta \psi...
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### I Why is linear momentum not conserved or realistic here?

- This is the situation provided where it has to be demonstrated that linear momentum is not conserved or unrealistic. My assumptions are: there's no friction due to how Godzilla is moving, the sand is acting as a lubricant. Velocity is constant. This is an elastic collision. I also guess that...
11. ### I Linear Program:Multiple Optima for multivariable Obj. Func.?

I know there can be an infinite number of solutions when the objective function with 2 variables has an equal slope as a constraint's slope (assuming the constraint is affecting the feasible region and not a redundant constraint). How can you know there are multiple optimal solutions for...
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### Linear and Non-linear Equations (QM)

1. Problem Recall that we defined linear equations as those whose solutions can be superposed to find more solutions. Which of the following differential/integral equations are linear equations for the function u(x,t)? Below, a and b are constants, c is the speed of light, and f(x,t) is an...
13. ### A R1^2 + R2^2 = R3^2?

Hey. I am working with OLS regression. First I run 2 regression operations with each having just one independent variable. Then I run another regression using both the independent variables from the first two regressions. If the explanatory "power" (R^2) in the third regression was to be the sum...
14. ### How stretching a rubber band, affects the range of a shot?

1. Homework Statement I did an experiment on how stretching a rubber band, affects the range or distance of a projectile shot. Variable list: Independent Variable Amount of stretching Dependent Variable Range or distance I created my catapults on my own and used two different rubbers...
15. ### I F-test longitudinal data

Can i Use a standard F-test on longitudinal data for a linear multiple regression? Mons
16. ### I Panel study, multiple linear regression, assumptions

Hey. I am doing a project where I am studying a set of companies over a 7-year period. I am doing a multiple linear regression analysis either with fixed or random effects (so, it's a panel study). What I am wondering is if the general assumptions/requirements apply when using the fixed/random...
17. ### Solving a fifth order non-homogeneous differential equation

1. Homework Statement Find the general solution of y^{(5)}-y(1)=x 3. The Attempt at a Solution I found the complementary function by substitution of the solution form y=e^{kx} giving k=0,1,-1,i,-i, so y_{cf}=a_0+a_1e^x+a_2e^{-x}+a_3e^{ix}+a_4e^{-ix} Now for the particular integral, the...
18. ### Proving basic linear ODE results

1. Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE...
19. ### Engineering Solving a linear circuit #2

1. Homework Statement This is the second problem from our test preparation cycle and i partially solved and understood it. The circuit in the picture is in the state of switch being open (STATE 1: SWITCH OPEN). Calculate $E1$ so that the change in the electrical energy of the capacitor is...
20. ### Engineering Solving a linear circuit

1. Homework Statement This is one of the problem i had had some help in solving but some things remain unclear to me and i will highlight them by posting the exact question in the brackets next to the line of solution so that you can see what bothers me. Hopefully you can help me better...
21. ### Show: Elementary row operations don't affect solution sets

1. Homework Statement Show that elementary row operations don't affect solutions sets in linear systems 2. Homework Equations - 3. The Attempt at a Solution It's pretty easy to come up with a random linear system and perform ERO on them and showing that solutions are not affected, but is...
22. ### Engineering Solving a linear circuit

1. Homework Statement In the circuit above $E_1=E_2=E_3$, $Ig_1=Ig_2=Ig_3$ and $R_1=R_2=R_3=R_4$. In the case when the switched is closed the power of the ideal generator $E_2$ equals $P_{E_2}=-30W$. Then the switch opens. Calculate the $ΔP_{E_1}$. 2. Homework Equations 3. The...
23. ### I Linearizing a System of ODE's

I am given the equations of Lorenz with respect to deterministic non-periodic flow: \frac{dX}{dt} = Pr(Y-X), X(0)=X_{0} \frac{dY}{dt} = -XZ + rX - Y, Y(0) = Y_{0} \frac{dZ}{dt} = XY-bZ, Z(0) = Z_{0} where Pr is the Prandtl number, r = Ra/Rac is the ratio of the Rayleigh number to its...
24. ### 2nd Order Linear ODE-Derivation of system-issue

1. Homework Statement How exactly they combined equation1 and equation2 and got that system? I don't get that part. 2. Homework Equations A*(dy/dt)= -k*y eq1 A*(dz/dt)=ky-kz eq2 3. The Attempt at a Solution I tried substituting the 1st ky in the 2nd equation and then differentiating but...
25. ### B Angular Velocity in Linear Momentum Equation

m1v1 = m2v2 v = rω m1(rω)1 = m2(rω)2 m1ω1 = m2ω2 Does this make sense?
26. ### Question about conservation of linear momentum

1. Homework Statement [pardon my crude drawing] Say I have a ball that hits the a rod at rest pivoted around hinge A. The ball in the diagram is the solid black, and the rod is the thin line in the diagram and is rotating around the square-shaped hinge at the bottom. I am not an artist but...
27. ### Linear Programming Case Study - Case Problem

1. Homework Statement Linear Programming Case Study - Case Problem ( Page # 109 Decision making methods) “The Possibility” Restaurant? In the case problem, Angela and Zooey wanted to develop a linear programming model to help determine the number of beef and fish meals they should prepare each...
28. ### All possible planes, given two points

1. Homework Statement Find the equation of all planes containing the points P(2, -1, 1) and Q(1, 0, 0) 2. Homework Equations 3. The Attempt at a Solution I use PQ to get a vector, (-1, -1, 1). I some how need to use another vector so I can use the cross product to find the planes. So i...
29. ### -66 over -99 into a graph?

Okay, so I'm down to the last equation. -12/11 x 10 + 54/11 I get -66/-99. Is this right? If so how do I put it into the graph. -12/11 x 10 = -120/110 + 54/11 = -66/99 (I think I've went wrong somewhere)