First order Definition and 546 Threads

This is regarding the Bertrand theorem in the book Classical Mechanics by Goldstein. It is said that for more than first order deviations from circularity the orbits are closed only for inverse square law and hooke's law. What does first order deviation mean ?

First, some may ask why would do we care that we can convert a 3rd order or higher ODE into a system of equations? Well there are quite a few reasons. 1- Almost all first order systems are easier to solve numerically using computer systems (matlab, maple, etc). Yes, it takes some working out...

So this is an example given to me in one of the guides. I am somewhat confused as to why R and R1 are considered parallel when the switch is closed at t=0. I thought that the two resistors cannot be parallel since there is a voltage source between them. One hypothesis I had was to treat it...

I've seen this done in a video but I can no longer find the video! :( What I would like to do is approximate the solutions to a polynomial equation in terms of a small perturbation. For example, say we have y = f(x) and know the corresponding zeros exactly. How would I go about finding first...

I have a small problem with the first order logic, in particular, predicate logic Let us take this sentence as an example: Each teacher has given a form to each student. From this sentence, can we have different reading? This is my try to solve such problem, I did not know if this is the...

Hello I would like to check my reasoning about solutions of first order PDE. I've spell out (almost) all details. I'll consider the following problem: find ##u=u(t,x)## s.t. : $$ \partial_t u(t,x) + a(x) \cdot \nabla u(x) =0 \qquad \qquad u(0,x) = u_0(x)$$ say with smooth coefficient and...

Homework Statement I have this set of equation: My''+Cy'+Ky=0 but C=0 M is a matrix consist of {(-m) (0)/( -1/12mb^2) (-1/12mb^3)} and K is a matrix of {(-K1-K2) (-K2b)/ ((K1b-K2b)/(2)) (-K2b^2/2)} and y is a coordinate system which is (x1,θ) Now i have to convert these...

Homework Statement I'm trying to evaluate the following integral to calculate a first-order correction: $$\int_0^\infty R_{nl}(r)^* \delta \hat {\mathbf H} R_{nl}(r) r^2 dr$$ The problem states that ##b## is small compared to the Bohr radius ##a_o## Homework Equations I've been given...

In lectures, I learned that in first order perturbation, \hat{H}_0 term cancels with E_0 term because \hat{H}_0 is Hermitian. What property does Hermitian operators hold that cancels with the unperturbed energy?

Homework Statement The harmonic oscillator's equation of motion is: x'' + 2βx' + ω02x = f with the forcing of the form f(t) = f0sin(ωt)The Attempt at a Solution So I got: X1 = x X1' = x' = X2 X2 = x' X2' = x'' ∴ X2' = -2βX2 - ω02X1 + sin(ωt) The function f(t) is making me doubt this answer...

I'm having trouble solving the differential equation (a-bx)y'+(c-dx)y-e=0 with a,b,c,d,e constants. I tried laplace transforming it, but then I end up with yet another differential equation in the laplace domain because of the xy and xy' terms.

Hi, The problem is to solve: dy/dx = −[2x + ln(y)]*(y/x) Attempt: I have tried to see if it is exact, I found it not to be, I can't easily find a function to multiply by to make it exact either (unless I am missing something obvious). It clearly isn't seperable, nor is it homogenous (I know...

Homework Statement \frac{dy}{dx}\:+\:ycosx\:=\:5cosx I get two solutions for y however only one of them is correct according to my online homework (see attempt at solution) Homework Equations y(0) = 7 is initial condition The Attempt at a Solution \int \:\frac{1}{5-y}dy\:=\:\int...

Given are the following predicate symbols: Member(x) : x is a member of the bicycle club Chairman(x) : x is the chairman of the bicycle club Bicycle(x) : x is a bicycle Brand(x, y) : the brand of x is y Owns(x, y) : x is the owner of y a. Statement: every member of the bicycle club has the same...

Homework Statement So our lecture introduced first order circuits to us. We are trying to solve for v(t) in the following circuit. However in the notes he gets it into a form in which we can solve for v(t). However I can't seem to get it into the same form he has. This is more of a maths...

Homework Statement The ground state energy of the 1D harmonic oscillator with angular frequency ##\omega## is ##E_0 = \frac{\hbar \omega}{2}##. The angular frequency is perturbed by a small amount ##\delta \omega##. Use first order perturbation theory to estimate the ground state energy of the...

I have problems with question b. I don't know where to start. If the switch is on steady state, the current through capacitor would be 0 and i can't have K. This makes me confused. This is the Total response function of a): V0(t) = K^(-1/2t)

Homework Statement What's the first order term in the expansion ln(x) about x = 1? Homework Equations Taylor series formula The Attempt at a Solution The question is multiple choice, and the choices are x, 2x, or (1/2)x. However, when I calculate the first order term in the expansion of ln(x)...

what is a good book to learn first order differential equations ??

what is a good book to learn first order differential equations ??

Homework Statement Find a solution of $$\frac{1}{x^2}\frac{\partial u(x,y)}{\partial x}+\frac{1}{y^3}\frac{\partial u(x,y)}{\partial y}=0$$ Which satisfies the condition ##\frac{\partial u(x,y)}{\partial x}\big |_{y=0}=x^3## for all ##x##. The Attempt at a Solution I get the following...

Homework Statement Assume that there is a deviation from Coulomb’s law at very small distances, the Coulomb potential energy between an electron and proton is given by V_{mod}(r)=\begin{cases} -\frac{q^{2}}{4\pi\varepsilon_{0}}\frac{b}{r^{2}} & 0<r\leq b\\...

Homework Statement Use the Chain Rule to find the 1. order partial derivatives of g(s,t)=f(s,u(s,t),v(s,t)) where u(s,t) = st & v(s,t)=s+t The answer should be expressed in terms of s & t only. I find the partial derivatives difficult enough and now there is no numbers in the problem, which...

Homework Statement A process can be represented by the first order equation (4δy(t)/δt) + y(t) = 3u(t) Assume the initial state is steady (y = 0 at t = –0). (a) Determine the transfer function of this process in the s domain. (b) If the input is a ramp change in u(t) = 4t, determine the...

I need to solve the well known momentum equation in 3D cylindrical coordinates: ρ(∂v/∂t +(v.∇)v)=A where A and the velocity v are both local vector variables. I am actually looking for the stationary solution to the equation, i.e. no ∂/∂t term) I have tried evolving the velocity and tried...

Homework Statement [/B] Particle is moving in 2D harmonic potential with Hamiltonian: H_0 = \frac{1}{2m} (p_x^2+p_y^2)+ \frac{1}{2}m \omega^2 (x^2+4y^2) a) Find eigenvalues, eigenfunctions and degeneracy of ground, first and second excited state. b) How does \Delta H = \lambda x^2y split...

Homework Statement dy/dx = (x +y) / (x-y) , i am asked to find the first order differential equation , but the ans i gt is different from the ans given Homework EquationsThe Attempt at a Solution

The question is as follows: Suppose you find an implicit solution y(t) to a first order ODE by finding a function H(y, t) such that H(y(t), t) = 0 for all t in the domain. Suppose your friend tries to solve the same ODE and comes up with a different function F(y, t) such that F(y(t), t) = 0 for...

Homework Statement :[/B] I would like to know if the definition of the ZSR response means that the initial condition at any t0 needs to be 0.Homework Equations :[/B] let's say : we're trying to calculate the ZSR response from a first order equation of the voltage of a capacitor and the initial...

Homework Statement Solve the differential equation: dy/dx = 2/(x+e^y) Homework EquationsThe Attempt at a Solution I tried to use the substitution v=x+e^y, but I didn't get very far: v’=1+e^y y’ v’-1=(v-x)y' y’ = (v’-1)/(v-x) (v’-1)/(v-x) (x+v-x)=2 V (v’-1)/(v-x)=2 vv’-v=2(v-x) vv’-3v=-2x...

Basically, I am confused by one question in a practice paper in which the equation is given as follows: dy/dx = e^-2y and I know the general solution is equal to : y = -0.5e^-2y + C which would make sense if it was direct integration however it seems to me it is in fact separable...

Homework Statement We have a driven pendulum described by the following differential equation: \frac{d^2\theta}{dt^2} = \frac{-g}{l}\sin(\theta) + C\cos(\theta)\sin(\Omega t) I need to turn this second order differential equation into a system of first order differential equations (then...

My question applies to the case when the switch opens. By applying KCL in order to get a first order diff equation, the following problem arises when I choose different current directions (which shouldn't happen because KCL says the current direction doesn't matter because it will be fixed...

Homework Statement It asks me to find io(t=0-), io(t=0+), and Vc(t=0-). C=100μF R= 2kΩ Homework Equations V=I*R, i(t)= i(∞)+[i(0+)-i(∞)]*e-t/τ, Vc(0-)=Vc(0+) The Attempt at a Solution [/B]I first tried to calculate Vc(0-) as it will be the same as Vc(0+), stating that at t=0- the...

I thought I understood how to solve these sorts of equations, but apparently not.. 1. Homework Statement In Linear Algebra I'm solving diff eqs with eigenvectors to get all the combinations that will solve for a diff eq. The text then asked me to check my answer by going back and solving...

Homework Statement In a gas pipeline a mixture of hydrogen in air is transported. Usually, the hydrogen content of the gas is 1 volume %. It is well below the lower explosive limit as a hydrogen / air mixture is 4 % by volume hydrogen. gas pipeline is positioned a detector that activates an...

Hey! :o I want to check if there is a solution of a linear differential equation of first order in the ring of exponential sums $\text{EXP}(\mathbb{C})$. I have done the following: The general linear differential equation of first order is $$ax'(z)+bx(z)=y(z) \tag{*}$$ where $x,y \in...

Godels Incompleteness Theorem states that for any formal system with finitely recursive axioms we can construct a Godel sentence G that is unprovable within that system but is none the less true. Does this still apply to formal systems which, instead of creating Godel numbers for arithmetical...

Homework Statement We have the equation ## (\frac{dr}{ds})^2+(\frac{l}{r})^2=1 ## and want to solve to get ## r=\sqrt{l^2+(s-s_0)^2}## Homework EquationsThe Attempt at a Solution I have worked backwards, plugging in the solution to prove that it is correct, but the closest I have gotten to...

Hey there! So I have this circuit: with R= 1kOhm and L=10mH and for some reason the theoretical and real values differ drasticly for lower frequencies Brown lines are representing the asymptotic and approximation lines. Blue lines are connecting my measured values. Amplitude is the amplitude...

Could anybody explain me why indeed we can express the first-order correction to the n-th wave function \psi_{n}^{1} by linear combination \sum_{m} c_{m}^{(n)}\psi_{m}^{o}

I have an extremely messy system of differential equations. Can anyone offer any ideas for a general solution? p(t) is a function of t, and A is a constant.

Why in first order logic theories are not possible a demonstration with infinite steps?

Hi, I have a first order linear DE that I need to find the general solution for. I thought that I had, but my solution does not make sense when plugged back into the equation. I think that my method of separation of variables might be inapplicable here, but don't know the reason for this. I...

I want to convert this linear second order general form PDE to two equations: ##ϕ_{xx}+bϕ_{xy}+cϕ_{yy}+dϕ_x+eϕ_y+fϕ=g(x,y)## Converted equations: ##a_1 u_x+b_1 u_y+c_1 v_x+d_1 v_y=f_1## ##a_2 u_x+b_2 u_y+c_2 v_x+d_2 v_y=f_2## I want to find parametric values of ##a_1 ...f_2## How can I do...

Homework Statement I am attempting to understand this example shown below: Homework Equations During stead state DC, the capacitor is an open circuit and the inductor is short circuited. The Attempt at a Solution [/B] The questions I have are really related to the concepts as I don't...

Can anyone help me to solve a differential equation? I want to solve ∂v(p,t)/∂t=-p^2 v(p,t)-sqrt(2/pi)∫v(p,t)[1-δ(t)R(t)]dp+sqrt(2/pi)[δ(t)R^2(t) C] with initial data v(p,0)=0 where C is constant and the integration from zero to infinty Any suggestion please? Solution by volterra integral...

Hi. When using approximations such as g = n + h where h is much less than n ; what does " valid to first order in h" mean ? Does it mean include terms linear in h or does it mean drop those terms ? Thanks

Homework Statement ##y' = \frac{cos x}{sin y}## ##y' = \frac{6x^2}{y(1 + x^3)}## Homework EquationsThe Attempt at a Solution So I was working through some textbook problems and there's something about the solutions of the above equations I don't quite understand. The first one: ##\int sin...

Homework Statement What is the general solution of: y'=(3*y^2-x^2)/(2*x-y) Homework EquationsThe Attempt at a Solution This First Order equation is neither linear nor separable. I also have checked the Exact test, which turns to be Not Exact. Any help regarding how...