Differential equations Definition and 999 Threads

Hi all, I think this issue periodically resurfaces in PF. I have found a similar discussion in this closed post and possibly others. I'm posting this because I'd like to check my understanding, if anyone is available to provide some furtherinsight. So I'm trying to gather a "overall"...

Hello, I am studying control theory. And I have encountered something I have never considered or thought about. Consider a system with y as the output differential equation and u as the input. any(n) + ... + a1y(1) + a0y = bmu(m) + ... + b1u(1) + b0u Here, the subscripts indicate...

Homework Statement Given the following figure and the following variables and parameters, I have been able to come up with the set of differential equation below the image. My question is how does the system of equations 1 which I produced myself differ from the set of equations 2. Below I have...

Hello Guys, We haven't yet covered on how to solve 2nd order equation in class however we have this assignment given to us. Any tips would be appreciated for these 2 little problems. 1. Homework Statement We have this initial Equation: d2y/dt2−7dy/dt+ky=0, and we need to find the values of k...

I saw this problem and solved it but still I had one question... Homework Statement A rock falls through water with a continuously decreasing acceleration. Assume that the rock’s acceleration as a function of velocity has the form ay = g - bvy where b is a positive constant. (The +y direction...

Homework Statement Suppose the acceleration of a particle is a function of x, where a(x)=(2.0 s-2)*x. (a) If the velocity is zero when x= 1.0 m, what is the speed when x=3.0 m? (b) How long does it take the particle to travel from x=1.0 m to x=3.0 m. a(x)=(2.0 s-2) * x (a) V(x=3) = ? , V(x=1)...

Vibrations - Modelling system, equation of motion Hi, In the first question (question 4) in the attached file, how would you go about modelling the system and finding the equation of motion? All those masses are confusing me, I don't even know where to start. I don't know whether the angle...

Homework Statement Considering the function $$f(x) = e^{-x}, x>0$$ and $$f(-x) = f(x)$$. I am trying to find the Fourier integral representation of f(x). Homework Equations $$f(x) = \int_0^\infty \left( A(\alpha)\cos\alpha x +B(\alpha) \sin\alpha x\right) d\alpha$$ $$A(\alpha) =...

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...

Homework Statement Coupled Harmonic Oscillators. In this series of exercises you are asked to generalize the material on harmonic oscillators in Section 6.2 to the case where the oscillators are coupled. Suppose there are two masses m1 and m2 attached to springs and walls as shown in Figure...

Before I delve into this , I just wanted to know the basic approach. Do we look for symmetries because it gives us a systematic way to find coordinate changes that change the differential equation into a separable one? Thanks jf

Homework Statement Solve the following coupled differential equations by finding the eigenvectors and eigenvalues of the matrix and using it to calculate the matrix exponent: $$\frac{df}{dz}=i\delta f(z)+i\kappa b(z)$$ $$\frac{db}{dz}=-i\delta b(z)-i\kappa f(z)$$ In matrix form...

In driven SHM, we ignore an entire section of the solution to the differential equation claiming that it disappears once the system reaches a steady state. Can someone elaborate on this?

What is the most satisfactory explanation for guessing certain solutions to the differential equations encountered in damped & driven SHM?

Hi! Can anyone help me? If a constant number h of fish are harvested from a fishery per unit time, then a model for the population P(t) of the fishery at time t is given by: dP/dt = P(5-P) - h, P(0) = P0. a. Solve for the IVP if h = 4. b. Determine the value of P0 such that the fish...

Hi everybody. I'm using the Maple 13 software (in linux mint) to solve system compounded by the four below differential equations: > ode1 := (diff(m1(t), t)) = - m1(t) + (1/2)*tanh( m2(t) + m4(t) + cos(t) ); > ode2 := (diff(m2(t), t)) = - m2(t) + (1/2)*tanh( m1(t) + cos(t) ); > ode3 :=...

Hi at all, I've a curiosity about the role that the weight function w(t) she has, into the define of adjoint & s-adjoint op. It is relevant in physical applications or not ?

Hi everybody. I need to learn how to solve this kind of equation by decomposing y in a serie of functions. All the examples I have seen are of homogeneous functions. I would be extremely thankfull if someone pointed me to some text in which this is done-explained. Thanks for reading.

Homework Statement How does one show that q(t) is indeed a solution? Homework EquationsThe Attempt at a Solution My current idea is that i should come up with any form of solution, like q = Acos(ωt), and slot it in the RHS. Reason being that if q is indeed a solution, the result of the...

Homework Statement We can treat the following coupled system of differential equations as an eigenvalue problem: ## 2 \frac{dy_1}{dt} = 2f_1 - 3y_1 + y_2 ## ## 2\frac{dy_2}{dt} = 2f_2 + y_1 -3y_2 ## ## \frac{dy_3}{dt} = f_3 - 4y_3 ## where f1, f2 and f3 is a set of time-dependent sources, and...

Homework Statement Find all solutions of the given differential equations: ## \frac{dx}{dt} = \begin{bmatrix} 6 & -3 \\ 2 & 1 \end{bmatrix} x ## Homework EquationsThe Attempt at a Solution So, we just take the determinate of A-I##\lambda## and set it equal to 0 to get the eigenvalues of 3...

Homework Statement A cantilever of length ##L## is rigidly fixed at one end and is horizontal in the unstrainted position. If a load is added at the free end of the beam, the downward deflection, ##y##, at a distance, ##x##, along the beam satisfies the differential equation...

Firstly I know how to do this with first derivatives in differential equations - for example say we had ##\frac{dy}{dx}=4y^2-y##, and we're also told that ##y=1## when ##x=0##. ##\frac{dy}{dx}=4y^2-y## ##\frac{dx}{dy}=\frac{1}{4y^2-y}=\frac{1}{y\left(4y-1\right)}=\frac{4}{4y-1}-\frac{1}{y}##...

So, i am currently studying physics in a brazilian university. I am going to have a Calculus 2 course which, in Brazil, covers Ordinary Differential Equations and multi-variable differential calculus. So which challenging and rigourous books would you guys recommend for that? Thanks for the...

Hi, I need to solve a system of first order partial differential equations with complex variables given by What software should I use for solving this problem..? The system includes 13 differential equations ...

If the slope of the curve (derivative) at a given point is a number .

Homework Statement Homework Equations Circuit Equations. ##U_C=Q^2\2C## ##U_L=Li^2\2## The Attempt at a Solution For (a) I said ##100J## .But I think it might be ##200J## too.Here what I did; ##U_t=Q^2\2C## and I put ##Q=0.1C## and we know ##C##.Here I...

If you know phi it is about 1.618...=2cos36. The equations when x=phi which is equal to 0 is x^2-x-1=0. I took the first derivative squared and the second derivative cubed. The equation with x=phi is: [2x-1]^2+2^3=13 Check for yourself, if you fill in phi you get 13. Anyway, I do not know what...

Hi all, I'm wondering if anyone is able to point me in a direction regarding an aspect of stochastic differential equations. I have a situation in which I need to propagate a stochastic DE through time using measurement updates - however, the exact time at which each measurement arrives is...

I was trying to picture the third derivative of something Then i came across these ... What does displacement mean? The variable x is often used to represent the horizontal position. The variable y is often used to represent the vertical position Displacement=Delta x=xf-x0xf refers to the...

Homework Statement I'm fine with the first part. Part b) is causing me trouble http://imgur.com/xA9CG5G Homework EquationsThe Attempt at a Solution I tried subbing in the solution y1 into the given equation, but I'm not sure how to differentiate this, i thought of using integration by parts...

Homework Statement A submarine engine provides maximum constant force ##F## to propel it through the water. Assume that the magnitude of the resistive drag force of the water experienced by the submarine is ##kv##, where ##k## is the drag coefficient and ##v## is the instantaneous speed of...

I have this second order differential equation but I'm stumped as to how to solve this since the zeroth order term has a Sine function in it and the variable is embedded. ##\ddot y(t) + 3H (1+Q) \dot y(t) -m^2 f \sin(\frac{y(t)}{f}) = 0## ##H~##, ##~Q~##, ##~m~##, and ##~f~## are just...

Homework Statement Hey guys I'm struggling to find much information of modelling single species population dynamics that relates to this question. A question like this is going to be coming up in my final exam and I need to be able to solve it. I'm struggling to even know where to start. I'm...

I am trying to come up with an analytical solution (even as a infinite series etc.) for the following diffusion-convection problem. A thin layer of gel (assumed rectangular) is in direct contact with a liquid layer (perfusate) flowing with velocity v in the x direction (left to right) just...

Homework Statement Let f : I → C be a smooth complex valued function and t0 ∈ I fixed. (i) Show that the initial value problem z'(t) = f(t)z(t) z(t0) = z0 ∈ C has the unique solution z(t) = z0exp(∫f(s)ds) (where the integral runs from t0 to t. Hint : for uniqueness let w(t) be another...

Homework Statement [/B] There is a swing door with a damper. The characteristic polynomial (I have done it correctly) is: 0.5*r^2+1.5*r+0.625 General solution for x(0)=x_0 and v(0)=v_0 is (I have found it without a problem): (1.25*x_0+v_0/2)*e^(-0.5*t)+((v_0+0.5*x_0)/(-2))*e^(-2.5*t) Now the...

Hi, i have a question about a proof of some recursive equation, the function is $$c_{n}(a)=\int_{0}^{\pi } \frac{cos(nx)-cos(na)}{cos(x)-cos(a)}$$ whit ##n\in \mathbb{N}## and ##a\in \mathbb{R}## . whit some algebra is easy to see ##c_{0}(a)=0## and ##c_{1}(a)=\pi## and the recursive...

For what differential equations would having much quicker or financially cheaper methods of solving them significantly benefit scientists or engineers?

Imagine I have a complicated second-order differential equation that I strongly suspect can be derived from a Hamiltonian (with additional momentum dependence beyond p2/2m, so the momentum is not simply mv, but I don't know what it is). Are there any ways to test whether or not the given...

Hi guys. I am currently stuck on the classic snow plow problem. I have the following differential equation and initial conditions: @ 7am plow starts off to clear snow at a constant rate By 8am, plow has gone 4mi By 9am, plow has gone an additional 3mi Let t=0 when it started to snow, when did...

Homework Statement This is new for me, so forgive me my clumsiness. I am working on the following problem: A particle p is moving with a velocity v1 = c (speed of light) towards an object q, which is moving in the same direction with the speed v2, where v1>v2. Now, v2 is a function of the...

Homework Statement Ok, so I am attempting to solve a projectile motion problem involving air resistance that requires me to find the total x-distance the projectile traverses before landing again. Given: \\ m=0.7\text{kg} \\ k=0.01 \frac{\text{kg}}{\text{m}} \\ \theta=30 \degree Homework...

From the equation of motion of inflation, $$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + \frac{dV}{d\phi} = 0$$ Example: ##V= \frac{1}{2}m^2\phi^2## $$\frac{d^2\phi}{dt^2} + 3H\frac{d\phi}{dt} + m^2\phi = 0$$ If I want to make the DE dimensionless then I let ##~t = \frac{1}{H_o} \tilde t~## and...

Homework Statement So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show...

Homework Statement dv/dt = 9.8 - (v/5) , v(0) = 0 (a) The time it must elapse for the objet to reach 98% of its limiting velocity (b) How far does the object fall in the time found in part (a)? Homework Equations (dv/dt)/(9.8-(v/5)) The Attempt at a Solution I'm a little overwhelmed by this...

From cosmology, the tensor to scalar ratio is ##r=16\epsilon## where ##\epsilon=-\frac{\dot H}{H^2}## is the Hubble slow roll parameter. From warm inflation, $$\ddot \phi + (3H+\Gamma)\dot \phi + V_\phi = 0 ,\quad H^2 = \frac{1}{3M_p^2} (\frac{1}{2} \dot \phi^2 + V)$$ where ##H## is the Hubble...

Hello everyone. I'm an undergrad physics major with one semester left and I'm having some trouble. I took off 3 years to work on my depression and came back last spring to finish my senior year. Now, before I left I was struggling in all my classes due to my depression, but one was worse...

Homework Statement Find the values of the constant a for which the problem y''(t)+ay(t)=y(t+π), t∈ℝ, has a solution with period 2π which is not identically zero. Also determine all such solutions Homework Equations With help of Fourier series I know that : Cn(y''(t))= -n2*Cn(y(t)) Cn(y(t+π)) =...

Homework Statement For a following differential equation d^2y/dx^2-4y=(e^x)/x  Find the solution using numerical methods Homework Equations d^2y/dx^2-4y=(e^x)/x The Attempt at a Solution %num dx=0.01; x=1:dx:3; l=zeros(1,length(x)); m=zeros(1,length(x)); l(1)=1; m(1)=0.25; for...