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

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How t find z (unknown) in Runge-Kutta question

I'm trying to solve the below question solve $\dfrac{dx}{dy}=\dfrac{1}{x+y}$ for $x=0.5$ to $z$ using R-K (order $4$) with $x_0=0$, $y_0=1$ (take $h=0.5$). Please help me and tell me how to ...
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Blow up solution of a Riccati's equation.

Consider the Cauchy problem $$ \left\{ \begin{array}{l} \dot x=x(t)^2+t\\ x(0)=0 \end{array} \right. $$ Show that its solution is not defined in $[0,3]$.
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Function satisfies differential equation.

Given the D'Alembert operator D'Alembertian $\Box$, I want to show that $$ G(x,t,x_0,t_0):= \frac{\delta \left(t_0 + \frac{||x-x_0||}{c} -t \right)}{||x-x_0||} $$ satisfies $$ \Box G(x,t,x_0,t_0) = ...
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solution of 3rd order non linear differential equation

I need help regarding solution of this equation which has been solved in a research paper but I cant figure it out. Please help $f^{\prime\prime\prime} + 3ff^{\prime\prime}- 3(f^{\prime})^2 - (Ha +\...
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327 views

Finite Difference - Forward Difference with 2nd order Accuracy: What to do at the boundary?

I implemented a BVP using a first-order finite difference scheme after the shooting method did not work reliably. Its the first time I have worked with this. The code works but I would like to move to ...
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I need to model a population where every component have a fixed life span.

I have a certain population, which growth is function of a certain factor. I have already modeled the growth. Now I need to impose that every individual in the population have a fixed life length. Can ...
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27 views

Computer Code Friendly Books On Differential Equations?

When I need a differential equation for this or that application I generally search (by hand) through old paper and ink books written by mechanical or electronical (electrical) engineers. Sometimes my ...
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50 views

Find the equilibrium of the given ODE

Given the following non homogeneous, autonomous ODE: $$y'(t) = Ay(t)+b $$ with $y(0) = y_0$ where $y_0 \in R^{2m}$ is a known vector. Also , $A$ is $2m$ by $2m$ and invertible, and b $\in R^{2m}$. ...
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64 views

Uniqueness of solution for a system of differential equations

A friend of mine working on Auction Theory needs to establish uniqueness of solution (up to initial and boundary conditions) of a system of differential equations of the form $$ F(y_1,y_2,y_3,\dot{y}...
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61 views

Discretization of an Euler-Bernoulli

Given the following Euler-Bernoulli equation: $$ (s(x) w(x)'')''= q(x),\ \ x \in [0,1]$$ Could someone explain why the following discretization scheme may not be a good idea? \begin{align*} (sw'')''...
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Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
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98 views

Second order, inhomogeneous, linear differential equation

I come across this equation in book $$F(z)=(1-\lambda + \mu )f(z) + (\lambda - \mu) zf'(z) + \lambda\mu z^2f''(z)$$ where $\lambda \not= 0$ and $\mu \not= 0.$ My question is how to find $f$? Can ...
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67 views

Converting a series to a recursive expression

Let $e_i$ be a unit vector with one 1 in the $i$-th element. Is the following expression has a recursive presentation? $$y = \sum_{k=0}^{\infty} {\frac{{{X^k} e_i}}{\|{{{X^k} e_i}\|}_2}} $$ where $X$...
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112 views

Uniform perturbative solutions to the Mathieu equation

The Mathieu equation is a second-order linear differential equation given by $$y''(t) + [a - 2q\cos(2t)]y(t) = 0$$ There are two special functions defined as linearly independent solutions to Mathieu'...
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30 views

Saari's homographic conjecture and the actual definition of homography

By the wikipedia definitions found here and here, and especially by the definition implicit in this MSE post, it seems two images are homographic if they are renderings of the same set of points in ...
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31 views

Help solving particular D.E

I'm going through past exams for revision and couldn't get the same answer as the markscheme for this problem. QP http://papers.xtremepapers.com/CIE/Cambridge%20International%20A%20and%20AS%20Level/...
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Equation of a curve with a local minimum fixed at $x=a$ when we rotate the curve about the origin.

We have a strangely curved plank. If we place a round weighted object on it, it will rest itself at one point of it, when we incline the plank slowly, the object will gradually move towards a resting ...
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206 views

Central manifold theorem => Stable/unstable manifold?

I'm a bit confused why we always separate the stable/unstable manifold theorem and the central manifold theorem. The stable/unstable manifold theorem applies to a hyperbolic point ($\mathrm{Re}(\...
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86 views

Verification of Fourier transformation of Io-sinh function

I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...
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133 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
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66 views

Stochastic differential equation for a Fokker-Planck-type equation with a non-derivative term

I have something similar to a Fokker-Planck equation of the form $\frac{\partial}{\partial t}f( x,t) = A(x,t)f(x,t)- \frac{\partial}{\partial x}[B(x,t) f(x,t)] +\frac{1}{2}\frac{\partial ^2}{\partial ...
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42 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial x^3}...
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Long-time asymptotic behaviour of a system of two ODEs

We have the following nonlinear ODE: $$ f' = af-bg -(f+g)^k \bigl(f'(0) +g'(0)\bigr) + f'(0), $$ $$ \bigl(G-T(x)\bigr) g' = -af+bg - g'(0), $$ where $a,b,k,G$ are constants, $f'(0)$ and $g'(0)$ are ...
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best introductory intuitive books for learning ODE

I want to know best introductory intuitive books for learning ODE (mainly interested in Picard' theorem, Gronowall's inequality and most importantly stability). I started with Philip Hartman. Not ...
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find matrix A given its exponential

I'm given $e^{At}$ and I need to find A From http://www.math24.net/method-of-matrix-exponential.html I see that $$\frac{d}{dt}(e^{At})=Ae^{At}$$ so does it mean, that to answer my question I just ...
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82 views

Mathieu equation solution with non-periodic boundary conditions

I need to solve the Mathieu equation: $y''(x)+(a-2q \cos(2x)) y(x) = 0$ but with the unusal boundary condition: $y(x+\pi) = e^{i \alpha}y(x) \quad , \quad \alpha \in R$ if $\alpha = 0$ than the ...
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72 views

second order ODE :- solution

We have $y''-Py'-Qy = 0 $ where P,Q are $P = K_1+K_2x, Q =K_2 $. $K_1,K_2$ are constants. y' means derivative with respect to x . Please suggest a solution for y. Thanks
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Removing parametrization from a system of equations

Consider the following system : $$ \begin{aligned} \frac{d^2t}{d\lambda^2} &= -f\left(t\right)\frac{d t}{d \lambda}\frac{d t}{d \lambda} -A\frac{d g\left(t,x\right)}{d \lambda}\frac{d t}{d \...
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Differential Equation Involving a Bivariate PDF and its Marginal CDF

I have a differential equation of the form $$ P(x_1,0) = R(x_1) Q(x_1), $$ where $P$ is an unknown, isotropic bivariate probability density function (pdf) i.e. $$ P(x_1,x_2) = P(x_3,x_4), \quad \...
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97 views

Laplace transform of Differential Equation with a piecewise function

Hi I have this question and I am horribly stuck at one part and I cant seem to figure out if i did something wrong so any advice or help would be greatly apprecaited. Here is the question: $$y''-y'-...
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The phase portrait of a second order of nonlinear system using matlab

I have the following system $$ \ddot{x} + 0.6\dot{x} + 3x + x^{2} = 0 $$ In the book I'm reading, the phase portrait of the nonlinear system for the aforementioned equation is I would like to ...
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Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
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Power series solution for a DE with Frobenius method

The given DE is $(x²-3)y"+2xy'=0$ Since there is a singular point ($x=\pm\sqrt{3}$) I used the Frobenius method. I found two indicial relationships: $-3r(r+1)=0$ and $-3(r+1)(r+2)=0$ because I have ...
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Solution of a DAE system of two ODE of second degree

I should solve the following DAE system: $$\ddot{x}(t)=-\alpha y(t)$$ $$\ddot{y}(t)=\beta x(t)$$ with the conditions: $x(t)\ge0$, $y(t)\ge0$ and $x(t)+y(t)=N$ with $N\gt 0$. I'm able to solve the ...
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56 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + a_1(x)...
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Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
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the initial mass m(0) of a population of fish is one thousand tons at the beginning of a year

the initial mass m(0) of a population of fish is one thousand tons at the beginning of a year the population , if left alone , increases its mass with a proportionality constant 0.1 per month. ...
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Do these inequalities make sense?

I have two sets of inequalities and i just want to know if they are correct. The parameters $\mu, K, d_1, \sigma_1,\sigma_2$ and dependent variables $H,F$are positive. Also $\sigma_2>\sigma_1$. \...
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94 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
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Uniqueness of holomorphic solutions of a differential equation

Given two polynomials $p,q\in\mathbb C[z]$ consider the initial value problem \begin{align*} f(z)-p(z)f'(z)&=f(z^2)-q(z)f'(z^2),\qquad z\in\mathbb D,\\ f(0)&=0, \\ f'(0)&=1. \end{align*} ...
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Simple differential equation modelling question.

The question is: A chemical dissolves in water at a rate equal to 10% of the amount of undissolved chemical per hour. At time $t$ hours the amount of undissolved chemicalis $x$ grams. Initially the ...
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System of ODEs and DAE system

Let us consider the following system of ODEs: $$ y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0 $$ and the following one: $$ y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0. $$ $f$ and $g$ ...
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A solid rocket model: a differential equations set with ending time unknown

I am modelling a rocket model. Consider a solid rocket motor, (let us for sake of simplicity assume that the propellant is distributed in the case with a cylindrical shape: see shape in fig.1 of the ...
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35 views

How to find this common solution?

I have the system $$ \ddot{x}=-\frac{k_1 x}{m},~~\ddot{y}=-\frac{k_2 y}{m} $$ with $k_1,k_2,m > 0$. How can I solve this? Can you give me a hint? Edit My idea is to write it this way: $$ \dot{\...
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$\gamma(t)$ is not asymptotically stable unless $\int_0^T \nabla \cdot f(\gamma(t))dt \leq 0$

Let $f \in C^1(E)$ where E is an open subset of $\mathbb{R^n}$ containing a periodic orbit $\gamma(t)$ of $x'=f(x)$ of period $T$. Then $\gamma(t)$ is not asymptotically stable unless $$\int_0^T \...
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37 views

Solving $u_{\xi\eta} = 0$ and differentiability conditions on solutions

After transformation someone often encounters the PDE $$ u_{\xi\eta} = 0 $$ but I am quite confused about the differentiability conditions of its solution (for example in this post I read different ...
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How to I find $y(t)$ when the flow rate in is $=r$ and concentration of chemical $Y$ coming in is $=X$ grams per liter?

A tank initially contains $1$ liter of water and $628$ grams of chemical $Y$. A solution containing $X$ grams per liter of chemical $Y$ flows into the tank at the rate of r liter/hour. The mixture ...
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30 views

Uniqueness of solutions to ODEs for functions of complex domain

Given the ivp $$ \tag{1} \frac{df}{dz}=F(z,f), \hspace{1cm} f(z_0)=f_0 \hspace{1.5cm} f:\Bbb{C}\rightarrow\Bbb{C} $$ where $f$ and $F$ are complex valued and analytic, then does it folow that $(1)$ ...
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46 views

Poisson differential equation

I'm stuck on an old exam question: Let $\Omega = \{(x,y) \in R^2 : 1 < x^2 + y^2 < a \}$. Determine the unique solution for the following boundary condition problem: $\Delta u = 1$ for $(x,y) ...
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76 views

Solving a system of linear ODEs

Based on my previous post, I have been stuck on this for a few hours now. I want to solve for $x$ and $y$ from the equation $$\frac{dx}{dt} + \frac{dy}{dt}=a-(b+c+d)y-bx.$$ The original two equations ...