Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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A property of solution of ODE $y''+p(x)y=0$

Let $f$ be a solution of the following equation $y''+q(x)y=0$, $q$ is continuous on $\mathbb{R}$ such that $q(x)\leq 0$ for all $x\in\mathbb{R}$. We have $f$ is defined on $[a,+\infty)$, ...
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Which operators other than self-adjoint operators have no purely imaginary eigenvalues?

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...
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multivaraible chain rule proof

I wanted to prove the multivaraible chain rule; I had to prove that $df \large \frac {({x(t)},{y(t)})}{dt} = \frac{∂ f}{∂ x}\cdot\frac{dx}{dt} + \frac{∂ f}{∂ y}\cdot\frac{dy}{dt}$ So, I took the ...
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ODE of the Inverse Function

Problem: Show that the autonomous ordinary differential equation $y'=F(y)$ has a unique solution with initial condition $y(t_0)=y_0$ in a neighborhood of $t_0$ provided F is continuous and ...
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Method of Undetermined Coefficients reasoning

So this method is used to find a particular solution for higher order non homogenous linear differential equations. My problem is, it involves guesswork and hasn't got a solid proof. My textbook tries ...
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44 views

Differential equation whose solution is Erlang distribution

I am working on a proof (Probability Density Question Involving an Integral Equation (from Karlin & Taylor's A First Course on Stochastic Processes)) and got stuck. Now I would like try ...
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13 views

WKB for a sixth order eigenvalue problem

I have the following 6th order eigenvalue problem: $$ (D^2 - \alpha^2)^3 y(x) = -\alpha^2 \lambda Q(x) \, y(x), \quad 0 < x < 1, \quad \text{+ BCs}, $$ where $D = \mathrm{d}/\mathrm{d} x $, ...
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Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
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Particular solution of a second order differential equation

I have a general (maybe naive) question. Let's suppose I have the following diff. eq.$$y''(x)+a~y'(x)=b(x)$$ To solve it, I cand do a change of variable and write: $$u'(x)+a~u(x)=b(x)$$ Then, if I ...
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Book on Ordinary Differential Eqnns

What is a complete book containing all the topics listed below : 1.Ordinary Differential Equations (ODEs): Existence and uniqueness of solutions of initial value problems for first order ordinary ...
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global manifolds

Can you also explain why the global stable manifold is the union of the flow of the local stable manifolds for t < 0? Why do we not include all t?
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How to solve this ODE numerically?

I have a question about how to solve this ODE numerically: $$\frac{C}{4}y'^2+\frac{C}{4}y''y+(0.098)^2y''y'''=0$$ where $C$ is a constant and the initial conditions are $y(0)=y''(0)=0$ and $y'(0)=1$. ...
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ODE with with translated arguments

I'm working on a problem of Approximation Theory and to move foward I have to solve the following ODEs: \begin{equation} -2\pi iw.e^{-2\pi itw} = y'(t) + y(t-\frac{1}{2}) \\ y(-1/2) = 0, \\ -2\pi ...
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Nondimensionalization of complex nonlinear ODE

I am interested in obtaining the nondimensional form of the rather complicated complex first order ODE $$\left(i-\frac{1}{2\Omega}f_{m,n}\right) \frac{d a_{m,n}(t)}{dt} =E_{m,n}^{\text{kin}}(t) + ...
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Deriving equation for sequential decay?

The differential equation describing the decay of a particle (p1) into another particle (p2), which then decays into a further particle (p3) is: where is the number of p2 particles, and is the ...
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22 views

Help solving a Differential Equation?

dx/dt= x-4y dy/dt=4x=7y I found the eigenvalue to be -3 and the general solution to be $x=e^{3t} x_{initial} +te^{3t}(-2x-7y)$ and $y= e^{3t} y_{initial}+ te^{3t} (-x+y)$ Determine the particular ...
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What is a “limit circle boundary condition”?

I came across the notion in an article on differential equations and I did not find any satisfactory answer on it as yet. In more detail, it was concerning the differential operator ...
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39 views

IVP to $y'=\frac{x(y-5)}{y^3-y}, y(2)=5$

Consider the following initial value problem: $y'=\frac{x(y-5)}{y^3-y}, y(2)=5$ a)Show that there is an unique solution b) Find a solution to this IVP and determine the maximal domain of definition ...
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29 views

Region of absolute stability

We have the problem $$\left\{\begin{matrix} y'=\lambda y &, t \in [0,+\infty), \lambda \in \mathbb{C}, Re(\lambda)<0 \\ y(0)=1 & \end{matrix}\right.$$ Applying the Backward Euler method ...
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Is there an english edition of Jorge Sotomayor's book on differential equations?

I am currently using "Lições de equações diferenciais ordinárias", in portuguese, by Jorge Sotomayor. However portuguese is not my best language by a long shot, and I struggle a little. Does anyone ...
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Determining phase portrait of the following systems of ODE

Given the following systems of ODE$$x'=x-3y$$ $$y'=4x-6y$$ Draw its phase portraits First i calculate the eigenvalue to be -3 and -2. Hence both eigenvalues are real and negative. And the phase ...
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Confusion about jump discontinuity and Green's function of a B.V.P.

What is the jump of a function with discontinuity of first kind ? I know that the jump of $f(x)$ is difference between left-hand-limit and right-hand-limit in modulus. That is $$jump ...
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How to solve this ODE: $\frac{dS}{dt}=\Pi-\beta SZ-\delta S$ $\frac{dZ}{dt}=\beta SZ+\zeta R-\alpha SZ$ $\frac{dR}{dt}=\delta S+\alpha SZ-\zeta R$

There is a way to solve this ordinary differential equations system analytically? $\frac{dS}{dt}=\Pi-\beta SZ-\delta S$ $\frac{dZ}{dt}=\beta SZ+\zeta R-\alpha SZ$ $\frac{dR}{dt}=\delta S+\alpha ...
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Stable distribution law related to the Dickman function

It is known that if $U$ is a random variable with uniform distribution on the interval $(0,1]$, then the random variable defined as $$ X\stackrel{d}{=} U^{1/\alpha}(1+X) $$ where $\alpha>0$ ...
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20 views

Perturbation of Ordinary Differential Equation Example

This is from Arnold's book on ODE's. Does anyone know of a reference or example where I could see how a linear equation arose in the way he mentions? Earlier he claims that linear ODEs are useful ...
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Stability of Difference Equations

Suppose $\dot{x}(t) = h(x(t))$ is a well posed ODE and $x^*$ denotes an asymptotically stable equilibrium point. Let $B$ be an open set containing $x^*.$ Also suppose that $B$ lies in the region of ...
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Why would you choose the Method of Frobenius over a Power Series solution to solve a DE?

I'm trying to determine where it would be more appropriate to use one or the other. To further clarify: Where would it make more sense to use: $y=\sum_{n=0}^{\infty}c_n(x-x_0)^{n+r}$ instead of ...
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28 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
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Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
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Homogeneous Dilation of the Domain in the Free Membrane Problem

Consider the Neumann boundary value problem of the Laplace operator: $$ \begin{cases} \Delta u+\mu u=0,&\text{in }D,\\ \frac{\partial u}{\partial n}=0,&\text{on }\partial D. \end{cases} $$ Let ...
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Find the minimizer of the functional

Find the minimizer of the functional $ l= \int u(t) $ with $u(1)=u(1)=0 $ subject to $g=\int $$\sqrt{1+u'(t)} dt $ I want to solve it using E-L equation first $l^*=l- \lambda g$ then i used e-l ...
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basic reproduction number of a simple SEIR-model

the normal SEIR-model is: $\begin{array}{rll} \displaystyle{\frac{dS}{dt}}&=\mu N -\mu S -\beta \frac{I}{N} S & \text{Susceptible} \\ \displaystyle{\frac{dE}{dt}}&= \beta \frac{I}{N} S ...
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Stability of equilibriums

The question is to find the stability of the equilibriums of the system: $$\frac{dx}{dt}= 8x - 2y - 4x^3 - 2xy^2$$ $$\frac{dy}{dt}= x + 4y - 2y^3 -3x^2y$$ There are 3 equilibriums, $(0,0), (1,1), ...
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Is $x=0$ an ordinary or singular point? Two conflicting textbook solutions that use the same reasoning.

We're asked to determine whether $x=0$ is an ordinary point or singular point for the following two ODEs: $$\begin{align*}x y''+\sin x\,y&=0&(1)\\\\ x y''+(1-\cos ...
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Fourier Expansion of Hill's lunar problem

all! For my class I have to expand the following equation $y''(x)=4(\omega^2+q(x))y(x)$ in Fourier coefficients $y(x)=\frac{1}{2}y_0 + \sum^\infty_{n=1}y_n \cos(2nx)$ $q(x)=2\sum^\infty_{n=1}t_n ...
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Finding normalized eigenfunctions for $y'' + \lambda y = 0$

Find the normalized eignefunctions for $$y'' + \lambda y = 0$$ $$y(0)=0, y(\pi)-2y'(\pi)=0.$$ My teacher gives me this hints: Consider$$(py')'+qy+\lambda ry=0$$ where $p, p', q, r$ are ...
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Given a piecewise initial condition, how can the characteristic curve x be sketched when the solution x does not contain u terms?

The charac equation for x: $$\frac{\text{dx}}{\text{d$\tau $}}\text{=2t}$$ The solution x is $$x=t^2+x_0$$ Note that $$\tau=t$$ There is a problem. In order to sketch x, I require some ...
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Ordinary Differential Equations self-study reference request

I know there are a lot of reference requests for differential equations textbooks but none seem to be what I need. I'm looking for a book I can use for self study that isn't overly complicated and ...
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Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
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How to prove y component of the field is zero throughout the motion?

This is a pure mathematical question, here is a little background for the interested reader, you can jump directly to the mathematical part if you are not interested. background Imagine we have ...
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splitting a system of ODEs into linear constraints and a smaller system using matrix Null Space

This problem originates from chemistry. Let us assume we want to solve a system of ODEs describing the evolution of the concentrations of the species in a chemical system with n species and k kinetic ...
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Solve $y'' + \epsilon y' + 1 = 0$ with initial conditions $y(0) = 0$ and $y'(0) = 1$

Let $\epsilon << 1$. I guess I'm trying to use perturbation method but I've been getting really weird numbers when I'm determining the initial conditions. Can someone perhaps help me with ...
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Use a Lie series in order to find the solution to initial value problem

We were presented with a fairly difficult bonus question on my multivariable calculus exam today. I was hoping you all could hope me crack it. The question is as follows: Use a Lie series to find, ...
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Convergence of the Midpoint (Leapfrog) method when applied to $u'(t)=\lambda u(t)$?

So, I am trying to solve this question: where example 7.7 can be found here: http://i.stack.imgur.com/PVCIC.png My approach: Forward Euler (FE) method is given by: ...
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$(1+\cos x + 2\sin x)\dfrac{dy}{dx} + (\sin x-2\cos x)y = 5-3\cos x + 4\sin x$ case when $1+\cos x+2\sin x = 0$?

Find the general solution of $(1+\cos x +2\sin x)\dfrac{dy}{dx} + (\sin x-2\cos x)y = 5-3\cos x + 4\sin x$ I solved it, with dividing both sides by $(1+\cos x+2\sin x)^2$, Solution giving ...
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Solve a first order differential non linear equation

I have a first order differential nonlinear equation and I need some help to solve it. $\frac{dy}{dt}=a+ry+\frac{ry^2}{K}$ where a, r and K are constants to be found later. Any help would be ...
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Give the general solution to the fundamental set

$x^3y'''+2x^2y''=0$ I believe $\{1, x, \ln(x)\}$ is a fundamental set. Give it's general solution. Is it just $A+Bx+C\ln(x)$??
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Prove that there is at most one solution with Green's identity

Prove with the use of Green's identity that the boundary value problem $$\frac{\partial}{\partial{x}} \left( (1+x^2) ...
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How do I write this equation as a tridiagonal matrix to write the $n+1$ implicit formula?

I am doing a homework problem for my Applied Numerical Methods class, and I've worked the problem up to this point: $$ \large \frac{u_m^{n+1} - u_m^n}{k}=\frac{u_{m+1}^{n+1} - 2u_{m}^{n+1} + ...
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Solving a system of equations using operator method/notation

The system of differential equations is $$ \left( \begin{array}{ccc} x \\ y \\ \end{array} \right)' = \left( \begin{array}{ccc} 1 & -3 \\ 3 & 7 \\ \end{array} \right) \left( ...