Differential equations Definition and 999 Threads

Homework Statement Text from a classical mechanics textbook ( uploaded picture ) shows 2 diff equation describing the motion graphically presented in the uploaded picture. How were these set up? Homework EquationsThe Attempt at a Solution I don't have a slightest clue as how are these...

Homework Statement Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..) y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0 Homework Equations Lf'' = ((s^2)*F) - s*f(0) - f'(0) Lf' = sF - f(0) Lf = F(s) The...

Hey! :o Each element of the ring $\mathbb{C}[z, e^{\lambda z} \mid \lambda \in \mathbb{C}]$ is of the form $\displaystyle{\sum_{k=1}^n α_kz^{d_k}e^{β_kz}}$. A differential equation in this ring is of the form $$Ly = \sum_{k=0}^m \alpha_k y^{(k)}(z)=\sum_{l=1}^n C_lz^{d_l} e^{\beta_l z} , \ \...

As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...

Homework Statement I've attached an image of the problem question, it's Q1 I'm working on This is what I have so far: we have two components of SHM, position x and velocity v. when x = 0, v = a maximum, when v = 0, x = a maximum this is represented by sin & cos functions. where x =...

#17 If you can't see the picture: Suppose that y1, y2, and y3 are solutions to a third order constant coefficient homogeneous differential equation. Suppose further that for all real t, W(y1,y2)(t)>0, but also W(y1,y2,y3)(0)=0. Then there exists c1 and c2 such that c1y1(t) + c2y2(t) =y3(t) for...

There are two courses I can take for a Differential Equations class at my school. One is for Engineering students and is described this way (I'm a physics major fyi): This course presents an introduction to the theory of differential equations from an applied perspective. Topics include linear...

Hey guys, I have a question concerning the rewriting of a differential equation solution. In the example above, they rewrite [y=(plus/minus)e^c*sqrt(x^2+4)] as [y=C*sqrt(x^2+4)]. I understand that the general solution we get as a result represents all the possible functions, but if we were to...

Let the PDE $u_{xx}-4u_{xy}+4u_{yy}=0.$ Reduce to the canonical form.Good Morning MHB :). My problem is find the canonical form of the PDE know an variable change. But how I can transform the equation? Thanks.

Homework Statement Solve the initial value problem and determine at least approximately where the solution is valid (2x-y) + (2y-x)y' = 0 y(1) = 3 Homework EquationsThe Attempt at a Solution I know how to solve it, and I got the correct answer, which was: 7 = x^2 - yx + y^2 and then applying...

Homework Statement Given u(x.y), find the exact differential equation du = 0. What sort of curves are the solution curves u(x,y) = constant? (These are called the level curves of u). u = cos(x2 - y2)The Attempt at a Solution partial derivative du/dx = (-2x)sin(x2 - y2)dx partial derivative...

Homework Statement A simple electrical circuit consists of a voltage source E(t) = t*e-t volts, a resistor R = 1 and an inductor L = 1/10 H connected in series. It is assumed that I(0) = -10/81 a) The differential equation that governs the current I (t) in this circuit . b) Find the time...

Homework Statement Find the general solution to the equation. Homework Equations (dy/dx) - y - e^3x=0 The Attempt at a Solution [/B] I rewrote this as dy/dx - y = e^3x This is a linear first order ODE, in the form dy/dx + P(x)y = f(x) P(x) = 1; f(x) = e^3x The integrating factor =...

So I'm not sure whether this should be posted in the Physics forum or not, but here goes: I'm a Junior ME student at NC State taking a Vibrations course. We've gone over the general differential equation for mass-damper-spring systems where Meq⋅x'' + Ceq⋅x' + keq⋅x = Fextermal What I was...

Homework Statement [/B] I'm trying to 'solve' two coupled second order ODE's with the intent of putting them in state space. My specific problem is more complex and includes additional equations which are irrelevant. Essentially I can solve the problem if I know the solution to this. x1 and...

I have the following system of differential equations, for the functions ##A(r)## and ##B(r)##: ##A'-\frac{m}{r}A=(\epsilon+1)B## and ##-B' -\frac{m+1}{r}B=(\epsilon-1)A## ##m## and ##\epsilon## are constants, with ##\epsilon<1##. The functions ##A## and ##B## are the two components of a...

I have a question regarding the solutions to linear-ordinary differential equations. I had originally learned that the solutions to such differential equations consist of a homogenous solution and particular solution. The homogenous response is due to initial conditions while the particular...

Homework Statement Well I am looking for the particular integral of: d2y/dt2 + 4y = 5sin2t The attempt at a solution As f(t) = 5sin2t, the particular integral yPI should look like: yPI = Acos2t + Bsin2t dyPI/dt = -2Asin2t + 2Bcos2t d2yPI/dt2 = -4Acos2t - 4Bsin2t Subbing into the differential...

Homework Statement dy/dx + 2sin2pix = 0 -------Answer: y = 1/pi cos2pix + c Homework EquationsThe Attempt at a Solution I made several attempts but no success to the correct answer. The first step I made was subtracting 2sin2pix to both sides. I then used integration by parts, and this is...

I know there are a number of ways to do this problem, to increment the series etc. but, would someone please be able to explain how they get the answers for this problem simply and easily ? Thanks! A screen shot is attached

I'm having some difficulty with a problem from Boyce & DiPrima's Elementary Differential Equations and Boundary Value Problems, 9th Edition. The problem comes from Section 2.8: The Existence and Uniqueness Theorem and is part of a collection of problems intended to show that the sequence...

Let V1 be the voltage across C1 and V2 be the voltage across C2. I want to solve for V1 and V2 as a function of time. My idea was to use dV1/dt=I1/C1 and dV2/dt=I2/C2. Then using circuit rules i can express I1 and I2 as functions of V1 and V2 and substitute them into the previous diff eqs...

I am having a hard time understanding the conditions that set a plane to be called a region. According to definition 2.68, a set in the plane is called a region if it satisfies the following two conditions (p. 14): 1. "Each point of the set is the center of a circle whose entire interior...

I just finished up Stewart's Calculus Textbook, and the last section was on solving 2nd Order Non-Homogeneous Diff Eqs using power series. I've looked through Paul's Calculus page in the Differential Sections, and can see that there is still a lot more beyond Stewart's that I'd like to study...

Homework Statement solve y" - 2xy' + y = 0 Homework EquationsThe Attempt at a Solution in the worked example, the book gets from here: \sum\limits_{n=0}^{\infty} (n+1)(n+2)c_{n+2}x^n - \sum\limits_{n=1}^{\infty}2nc_nx^n + \sum\limits_{n=0}^{\infty}c_nx^n = 0 to here...

Homework Statement Homework EquationsThe Attempt at a Solution Any help would be appreciated

Homework Statement Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. For: y'' + 2y' + 10y = x^2e^{-x}\cos{3x} There's a modification performed and I'm not 100% confident as to why. Homework EquationsThe Attempt at a Solution The...

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...

Good evening people of PF! I have recently encountered a problem from Himmelblau's Basic Principles and Calculations in Chem. E. which asks to set up an energy balance for a tank, and then non-dimensionalize the differential equation before solving it. It's not the most complex task, but it's...

Textbook by Asmar. Would this class help me a lot for grad courses, like Jackson Electrodynamics or Sakurai Quantum? Debating to just finish up my upper levels and get As

During the summer, I plan on learning differential equations (ODE's and PDE's) from bottom to top, but I am unable to choose books due to a great variety present. Can you suggest books for me to read in the following order (you can add as many books in each section if you like);Ordinary...

Homework Statement Sorry to be bringing these in quick succession, but this one really has me perplexed. Is it possible that both the solution manual and I have two different but valid answers? I don't want to just go assuming that I'm right...but I think by subsuming certain parts of the...

Homework Statement Imgur link: http://i.imgur.com/C1epJzI.png My issues arise in part (b) but cause me to doubt my solution to part (a) as well. Homework EquationsThe Attempt at a Solution \frac{dx}{dt} = k(a-x)(b-x) \int \frac{1}{(a-x)(b-x)} dx = k \int dt Use partial fraction...

Homework Statement y'' + y =3*sin(2t) +t*cos(2t) Okay, so I have found the complimentary solution, and the first partial solution as listed in my work below. My problem is the work on the second partial solution. I have got all the derivatives plugged into the differential equation, my...

Hey! :o Is someone familiar with the following? We have linear differential equations with polynomial coefficients depending on x. $a_n(x)y^{(n)}+ \dots a_1(x)y^{(1)}+a_0(x)y^{(0)}=b(x)$ There are problems like if there are solutions, if the solutions are linear independent and so on and...

So there are these 4 differential equations problems that are ruining my day. I can't find any videos or notes for differential equations of calculus which is for some reason in my Calc 1 class. They seem kind of simple but I just can't seem to get started... any help is appreciated! 1b) When...

Homework Statement Solve the Differential Equation:[/B] When I take the partial derivative of each of these equations, I do indeed get that it is exact... However, when I do it the way my professor wants me to do it, I don't get the same result. He told us to multiply through by the common...

Homework Statement I believe I have solved this differential equation, yet do not know how the book arrived at it's answer... Solve the differential equation in its explicit solution form. The answer the book gives is... Homework Equations Separable Differential Equation The Attempt...

Homework Statement Solve the differential equation, explicitly. dy/dx = (2x)/(1+2y) The answer given by the book is... -1/2 + 1/2sqrt(2x - 2x^2 +4) Homework Equations Process for solving separable differential equations The Attempt at a Solution dy/dx = (2x)/(1+2y) (1 + 2y)*dy = 2x*dx...

Homework Statement [/B]Homework Equations The Attempt at a Solution I've highlighted two equations on the screenshot. How did it proceed from the first to the second? I'm actually confused with the absolute values. What is the idea behind getting rid of the first absolute value(1-5v^2) while...

Homework Statement dx/dt = (1+x2)et ; x(0) = 1 1/1+x2 dx/dt = et ∫ 1/1+x2 (dx/dt) * dt = ∫ et dt ∫ 1/(1+x2) dx = ∫ et dt arctan(x) = et + c so x(t) = tan (et+c) Homework EquationsThe Attempt at a Solution dx/dt = (1+x2)et ; x(0) = 1 1/1+x2 dx/dt = et ∫ 1/1+x2 (dx/dt) * dt = ∫ et dt ∫...

Hi, I have a wire that is fixed on one end and is under 2 lbs (approx) of tension on the other end due to a spring. Using a fixed/fixed closed form solution I found the first mode to be 73.6 Hz. In a vibration test, we found the first mode to be 110 Hz. Does this seem reasonable and where do...

Hello I have a system of differntial equations: dx/ds = sin(p) dy/ds=cos(p) dp/ds = k dk/ds = -1/EI(s)*(k*dEI/ds+f*sin(p)) x(0)=y(0)=p(0)=p(L)-pl = 0 These are nonlinear differential equations. I should use some sort of nonlinear finite difference. But I do struggle to setting up the finite...

I will try to explain this with an analogy. Let's have this equation: x^2 =9 And let's assume I don't know algebraic methods to solve it, so I create a list using excel with different values. And I see that if I put x=4 it doesn't work, if I put x=5 it is even worse and so on. But If I put...

Hello, I am a math major and I was wondering if you guys knew what would be a good rigorous differential equations text. I really like rigor (like Rudin analysis style rigor or whatnot), instead of the typical books that just focus on the method. I want the proofs and everything. I also would...

Hello, I wanted to study the behaviour of electrons in a spatially bounded system. I want to have a larger number of electrons, but I took 3 to start with and arrived at this system of coupled equations: \begin{align}\begin{bmatrix} \mathbf{\ddot{x_{1}}}\\ \\ \mathbf{\ddot{x_{2}}}\\ \\...

Homework Statement dv/dt = 9.8 - 0.196v Set in correct form: dv/dt + 0.196v = 9.8 Since p(t) = 0.196, u(t) the integration factor is given by: u(t) = e∫0.196 dt Multiply each term by u(t) and rearrange: (e∫0.196 dt)(dv/dt) + (0.196)(e∫0.196 dt)(v) = (9.8)(e∫0.196 dt) From now on we will set...

Homework Statement you are given a family of curves, in this case i was given a bunch of circles x^2+y^2=cx, sketch these curves for c=0,2,4,6, both positive and negative, solve the equation for c and differentiate both sides with respect to x and solve for dy/dx. You obtain an ODE in the form...

Homework Statement 1. y" + y = tanx, solve this DE. 2. dP/dt = P(1 - P) where P = (c1et / 1 + c1et), verify that P is a solution to this DE. 3. Given the pair of functionsx and y, show they solve this system: dx/dt = x + 3y dy/dt = 5x+3y x = e-2t + 3e6t y = -e-2t + 5e6tThe Attempt at a...

My school requires me to take a calc 3 course and differential equations for my major. I am scheduled right now to take differential equations in the summer, and calc 3 in the fall of this year. My school recommends they be concurrent classes, but I'm just too strapped down by my already almost...