Differential equation Definition and 1000 Threads

I'm currently studying the sensitivity of polynomial roots as a function of coefficient errors. Essentially, small coefficient errors of high order polynomials can lead to dramatic errors in root locations. Referring to the Wilkinson polynomial wikipedia page right...

I am given a modified SIR model in which the rate of decrease of susceptibles S is proportional to the number of susceptibles and the square-root of the number if infectives, I. If the number R of those who have been removed or recovered increases in proportion to the infectives, we have the...

Homework Statement Liquid is pouring into a container at a constant rate of 30cm^3s^-1 At time t seconds liquid is leaking from the container at a rate of 2/15 V cm^3s^-1, where V cm^3 is the volume of liquid in the container at that time. Show that -15 dV/dt = 2V - 450Homework Equations ...

ρCp (∂T/∂t) + k (∂2T/∂x2) = exp(-σt2)exp(-λx2)φo i have this equation... i was thinking of taylor series expansion to solve it and make it easier... any ideas on how to solve?

Homework Statement Homework Equations The Attempt at a Solution Because we are only looking at a cross section, I tried to reduce 5.3 down to just being a function of R and Theta. However I reasoned that there should be, based on this problem, no dependence on Theta either, so I figured I...

Homework Statement A steamer starts from rest, the engine exerting a constant propelling force Mf, where M is the mass of the steamer. The resistance of the steamer is assumed to vary as the square of the speed. Show that the distance x traveled in time t is: x = (V^2 /f) ln cosh (f*t/V )...

I'm going through the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman. On Page 138 (below) ) The authors take the derivative of a definite integral and end up with a definite integral plus another term. How did...

Homework Statement So they want me to obtain the general solution for this ODE. Homework Equations I have managed to turn it into d^2y/dx^2=(y/x)^2. The Attempt at a Solution My question is, can I simply make d^2y/dx^2 into (dy/dx)^2, cancel the power of 2 from both sides of the equation...

Homework Statement A mass M falls under gravity (force mg) through a liquid with decreasing viscosity so that the retarding force is -2mv/(1+t). If it starts from rest, what is the speed, acceleration, and distance fallen at time t=1. Homework Equations F=ma The Attempt at a Solution F =...

I am trying to solve this equation: d/dx[dF(x)/dx] = [c(c+1)/x^2)F(x), where c is a constant. Do I still use the characteristic equation to solve this? EDIT: Is it solvable using Dawson's integral rule?

Homework Statement Solve for the solution of the differential equation and use the method of variation of parameters. x`` - x = (e^t) + t Homework Equations [/B] W= (y2`y1)-(y2y1`) v1 = integral of ( g(t) (y1) ) / W v2 = integral of ( g(t) (y2) ) / W The Attempt at a Solution [/B] yc= c1...

Homework Statement dy/dx= 200-2y. y(0)=75 Homework EquationsThe Attempt at a Solution Do you move dx over and integrate. Do you just integrate it 200y-y^2+c

I just started learning Legendre Differential Equation. From what I learn the solutions to it is the Legendre polynomial. For the legendre DE, what is the l in it? Is it like a variable like y and x, just a different variable instead? Legendre Differential Equation: $$(1-x^2) \frac{d^2y}{dx^2}...

Hi all, I have derived a differential equation, which I don't know how to solve. I can do some numerical simulations, but would really be interested in, at least, knowing if an analytical solution exists, so would appreciate any help with it: (I have removed argument from y)...

Homework Statement A tank contains 60 kg of salt and 2000 L of water. A solution of a concentration 0.015 kg of salt per liter enters a tank at the rate 6 L/min. The solution is mixed and drains from the tank at the same rate. Find the amount of salt in kg at t = 3 hours Find the...

Homework Statement xy(dx)=(y2+x)dyHomework Equations integrating factor : u(x)=e∫p(x)dx standard form of linear DE: dy/dx + P(x)y=Q(x) standard form of bernoulli's differential equation: dy/dx + P(x)y=Q(x)yn change of variables v=y1-n The Attempt at a Solution xy(dy)=(y2+x)dx xy(dy/dx)=y2 +x...

The polynomial equation and it's private solution: $$(1)~~ay''+by'+cy=f(x)=kx^n,~~y=A_0x^n+A_1x^{n-1}+...+A$$ If i, for example, take ##f(x)=kx^3## i get, after substituting into (1), an expression like ##Ax^3+Bx^2+Cx+D## , but that doesn't equal ##kx^3##

d^2x/dt=0.01-0.01dx/dt =>x(t)=-100c1(e^-0.01t)+c2+t How do we find c1 and c2. Are they numbers or functions? d^2x/dt^2 instead of d^2x/dt gives the same solution, which means different c1 and c2

Homework Statement Solve the differential equation: (ex+1)cosy dy + ex(siny +1)dx=0 y(0)=3 Homework Equations none The Attempt at a Solution (ex+1)cosy dy + ex(siny +1)dx=0 (ex+1)cosy dy =- ex(siny +1)dx cosy/(siny+1)dy=-ex/(ex+1)dx ∫cosy/(siny+1)dy=-∫ex/(ex+1)dx using u sub on both the...

Homework Statement Solve each of the following differential equations: 4xydx + (x2 +1)dy=0Homework Equations None The Attempt at a Solution 4xydx + (x2 +1)dy=0 (x2 +1)dy=-4xydx dy/y=-(4xdx)/(x2 +1) ∫dy/y=∫-(4xdx)/(x2 +1) ln|y|=-2ln|x2+1| +C used u-sub on last step fo u=x2 +1

Homework Statement In this problem you will do numerical computer calculations. A skydiver of mass 75.0 kg jumps out of a plane at an altitude of 30.0 km above the surface of the Earth. His parachute fails to open. Assume there is no horizontal motion and the initial velocity is zero. We...

Homework Statement Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force: F=F_0+kv You could say mv = F0*t + kx, but the answer in the back of the book is an equation that is only in terms of x and v...

1. Homework Statement Posted Homework Equations Posted The Attempt at a Solution Posted I just need the work to be checked.

Homework Statement Write down the component form of the differential equations of motion of a projectile if the air resistance is proportional to the square of the speed. Are the equations seperated? Show that the x component of the velocity is given by \dot{x}=\dot{x}_0e^{^-\gamma s} where s...

Hello, I am currently taking ODE's and the class has an optional lab to accompany it. So far in the lab we've been doing some pretty basic stuff. But we've finally moved on to entering in differential equations, and I'm confused. 1. Homework Statement dydx+2x=2y How do I enter this equation...

Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...

Hello! Im trying to solve this second order differential equation: \begin{equation*} -\dfrac{d^2y}{dx^2}+\dfrac{3}{x}\dfrac{dy}{dx}+(x^2+gx^4+2)y=0 \end{equation*} Any idea? Maybe it could be converted to a Bessel-like equation (?) with an appropriate change of variables. The equation...

Hello - I'm not sure this is where this should go, but I'm working with Laplace Transforms and differential equations, so this seems as good a place as any. Also, I doubt this is graduate level math strictly speaking, but I went about as high as you can go in calculus and linear algebra during...

Homework Statement Homework Equations Leibniz notation: dy/dx = f(x) g(y) integral 1/g(y) dy = integral f(x) dx The Attempt at a Solution integral 1/y dy = integral sqrt (abs x) dx ln (y) = ? because sqrt (abs x) is not integrable at x =0 Then my thought is that y=0 is not unique

Homework Statement Solve the differential equation ##\displaystyle Cv^2 - mg = m\frac{d^2 y}{dt^2}## Homework EquationsThe Attempt at a Solution The problem is nonlinear, so we need to use unconventional methods. Specifically, if we can express the derivative of y with respect to v, then we...

Homework Statement Determine whether existence of at least one solution of the given initial value problem is guaranteed and, if so, whether uniqueness of the solution is guaranteed. dy/dx=y^(1/3); y(0)=0 Homework Equations Existence and Uniqueness of Solutions Theorem: Suppose that both...

Problem:Find the differential equation satisfied (i) by the equation of the family of tangents to y=x^2 and (ii) by the equation of the family of normals to y=x^2.

Homework Statement Find the values of m so that ##y = x^m## is a solution of ##x^2\frac{d^2y}{dx^2} - 3x\frac{dy}{dx} -12y = 0## Homework Equations ##y = x^m## ##y'=mx^{m-1}## ##y''=(m^2-m)x^{m-2}## The Attempt at a Solution So after plugging and chugging we get $$(m+2)(m-6)x^m = 0 $$...

Homework Statement The equation of motion of a particle is given by the differential equation ##\frac{d^2x}{dt^2} = -kx##, where ##x## is the displacement of the particle from the origin at time ##t##, and ##k## is a positive constant. 1. Show that ##x = A\cos{(kt)}+B\sin{(kt)}##, where ##A##...

Homework Statement [/B] (dy/dx)^2 = (1-y^2) / (1-x^2)Homework Equations Separating the variables I arrive with: dy/sqrt(1-y^2) = dx/sqrt(1-x^2) By integration on both sides by trigonometric substitution and putting it in a general solution: Arcsin y - Arcsin x = C The Attempt at a Solution If...

Hello folks, Let me explain this. Much effort, time, methods and books is devoted to the science of finding solutions of differential equations. But I cannot find anywhere the reverse problem. Basically I am thinking about the work that Maxwell did finding the differential equations that...

Homework Statement Homework EquationsThe Attempt at a Solution

Let us suppose that we are modeling a particle traversing some distance in space & that f(x)=x². Let the x-axis be time (hrs) & the y-axis be position in space (miles). What is the instantaneous velocity of said particle at t=4 hours? Let's assign variables to each coordinate in our x, y...

Hi, I was wondering if anyone had some advice on how to solve the following equation for ## F(b)##: $$ F(g(b)) h(b) + F'(b) s(b) - F(b)h(b) + h(b) + \int_{g(b)}^{b} v(x) F'(x) dx = 0 $$ Any hints on how to tackle this would be highly appreciated. Thank you!

Homework Statement A health club is opened, the fraction of members still enrolled t months from their initial visit is given by the function f(t)= e-t/20. the club initially accepts 300 members and will accept new members at a rate of 10 per month. How many people will be enrolled 15 months...

Hello I have a physics article who solve Poisson equation of the form: L2 ∂2f/∂x2=sinh(f) The proposed solution is: tanh(f/4)=exp(x/L) tanh(f0/4) with f0 a constant I suspect an error, something like a forgotten factor. How can I verify? (I tried but I failed) I forgot the limit...

Homework Statement Solve ##{dy/dx}-2xy=2x##Homework EquationsThe Attempt at a Solution Let ##P= -2x ## and Q= 2x, Integrating factor =## e^{-x^2} ## ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## ##y.e^{-x^2}={x^2} e^{-x^2}+∫ 2{x^3} e^{-x^2}dx## since ##y.e^{-x^2} = ∫ 2x.e^{-x^2} dx## then...

i was trying to learn differential equations ... i was lucky enough to find some good explanations for calculus and differential equations ... Mod note: unrelated image now deleted differential equations in this equation which is the dependent variables and which one is the independent...

Well, the problem says: From some height a object with mass m is thrown . Determinate the law that describes how the velocity of fall v changes, if on the object, besides gravity, acts the force air resistance, which is proportional to velocity v (the proportionality coefficient is k), ie must...

Hello I'm doing some problems in QM scattering regarding the Green's function. Homework Statement Determine the differential equation of G(\vec{r},\vec{r}',\omega) Homework Equations I've been given the Fourier transform for the case where the Hamiltonian is time independent...

Homework Statement [/B] Solve the differential equation Homework EquationsThe Attempt at a Solution [/B] I just can't integrate that (1+y)^(1/2)/(1+y^2)dy at the end... the other two integrals are trivial.

Homework Statement Homework EquationsThe Attempt at a Solution So the general solution is the sum of the null solution (Yn) and particular solution (Yp) I believe I just need to write: y = e2t + 5e8t - 5 + C and then find the derivative of both sides y' = 2e2t + 40e8t is this correct?

Suppose we have a differential equation with initial conditions ##y_{0}=y^{\prime}_{0}=0## and we need to solve it using a Green Function. Then we set up our differential equation with the right side "forcing function" as ##\delta(t^{\prime}-t)## (or with ##t^{\prime}## and ##t## switched I'm a...

Homework Statement The data given was the acceleration of the component over time, as below: time | Acceleration (m/s²) 0 | 0 0,2 |3,61 0,4 |4,5 0,6 |5,4 0,8 |7,508 1 |12 - The Rigidity constat is 12600000 [N/m] (for the spring) The damping constant is 6500 [N/m] (for the damp)...

Homework Statement Solve and obtain an expression for x in terms of θ given ## (3+ cos 2θ) dx/dθ = x sin 2θ## Homework EquationsThe Attempt at a Solution ...##dx/x = ((sin 2θ)/(3+ cos 2θ))dθ## , let ##u = (3+cos 2θ)⇒ du/dθ= -2 sin 2θ## thus ##∫dx/x = -1/2∫du/u ##...[/B]are my steps...