Tagged Questions

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

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0
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1answer
16 views

Sturm-Liouville problem

Find the eigenvalues and eigenfunctions of the the Sturm-Liouville problem $$(x^2v')'+\lambda v=0, \ 1<x<b$$ $$v(1)=v(b)=0, \ b>1.$$ The general solution is ...
1
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0answers
22 views

Related to Gronwall's Inequality.

The exercise is: Let $K \geq 0$, $f,g \geq 0$ continuous functions from $[a,b]$ to $\Bbb R$ and $x_0 \in ]a,b[$. Suppose that $f(x) \leq K + \left|\int_{x_0}^x f(t)g(t) \ \mathrm{d}t\right|,$ ...
3
votes
3answers
35 views

ODE $y'=ay+b/y$; no idea

I'd like to solve $$y'(t)=ay(t)+\frac{b}{y(t)}, \quad a,b\in\mathbb{R}$$ and have literally no clue how to begin. Additionaly the endpoint value is given by a transversalitiy condition like ...
0
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0answers
40 views

differential equation.

I'm stuck with this exercise. So, If someone might help me, I'll appreciate it too much. Let $U \subset \mathbb{R}^n$ be open and let $F:U\subset \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a smooth ...
0
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0answers
46 views

Asymptotic stability of $\dot{x}=-x^3$

So for an assignment I have to show that $\bar{x}=0$ is an asymptotically stable solution of $\dot{x}=-x^3 (\in\mathbb{R})$, using the definition of asymptotic stability (an equilibrium point $x^*$ is ...
0
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0answers
32 views

fundamental matrix function

I consider the homogeneous system $x'(t)=Ax(t)$ where $A$ is a $3x3$ matrix $$A=\pmatrix{-2 &0 &0 \\ 4 &-2& 0 \\ 1& 0& -2} $$ I need to determine the fundamental matrix ...
0
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1answer
17 views

Why is it that weak non-linearity reduces this equation in this manner?

I have this equation: $$\frac{d^2y}{dx^2}=\frac{dy}{dx}-2(3-y)y$$ which apparently has weak non-linearity and reduces to this: $$\frac{d^2y}{dx^2}=\frac{dy}{dx}-6y$$ It is not at all clear to me why ...
1
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2answers
40 views

How do I solve this differential equation? $y'=\frac{y}{x+y^3}$

So, hey. The equation is $y'=\frac{y}{x+y^3}$ So, I get something like this: $y'\left(x+y^3 \right)-y=0$, which I can't actually solve. I must admit I am slightly confused how to attack this one. What ...
0
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2answers
42 views

Non-existence of solution for $x'=g(x)$ if $x(0)=0$

Consider $$g(x) = \cases{ 1 & $x < 0$ \\ 2 & $x\geq0$}$$ and the differential equation $x'=g(x).$ Prove that there is no solution if we set $x(0)=0$. My idea is that the ...
1
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0answers
34 views

How to solve a discrete system of differential equations?

I am working with discrete ODE systems. I have a $(2x2)$ system approximation to the well-known predator prey model. In this $(2x2)$ approximation system, the right hand side of the system is given ...
1
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0answers
13 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
0
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2answers
18 views

Convergence of solutions for time to infinity in a 1D autonomous ODE

Given a constant $a > 0$, show that for every $y_0 > 0$ the solution of an initial value problem $$ \dot{y} = ay - y^5, \qquad y(0) = y_0 $$ satisfies $$ y_{\infty} := \lim_{t \to \infty} y(t) = ...
1
vote
1answer
31 views

Error in solution for a separable differential equation

I "solved" the differential equation $x'=x^2-1$ a couple of months ago, now I checked the solution with wolfram and it seem I was wrong... According to Wolfram the solution should be ...
1
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1answer
17 views

Row space of matrix

Hello! I am working on some differential equations homework and it is saying that my answer for this question is wrong, and i am not sure as to why. First I reduced the matrix A and then I read the ...
1
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0answers
11 views

2nd order linear differential equation with non-constant coefficients

Considering the equation $2y''+(x+1)y'+3y$ where $X_0=2$. Find the general term in each solution. That is, the general term for Y1,Y2 where $y=A_0(Y_1)+A_1(Y_2)$ I've solved this as ...
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0answers
6 views

Differentiational equation construct power series expansion

I got a question In order to improve the accuracy of your numerical estimate you are to use a power series expansion of y(x)to estimate y(1). (You may find it easier if you multiply both sides of ...
1
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1answer
40 views

Classify the fixed point at the origin of a dynamical system.

If we have a system $\dot x = -y+ax^3$ and $\dot y = x+ay^3$ I need to classify the fixed point at the origin for all real values of a. So I know we have to make the change of variables $ x = ...
1
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1answer
27 views

Solving a non-linear differential equation

Given the differential equation $$\ddot{y} + y + y^3 = 0$$ where $y = y(t)$, $\dot{y} = \frac{dy}{dt}$. By multiplying this equation by $\dot{y}$, assuming $\dot{y}>0$ and integrating, find ...
-3
votes
2answers
27 views

Time and work problems [on hold]

A and B can together finish a work in 30 days. They worked at it for 20 days and then B left. The remaining work was done by A alone in 20 more days. A alone can finish the work in?
0
votes
1answer
18 views

Differential equation using substitution?

How would I use the substitution $y = \frac{1}{\sqrt{v}}$ to solve the diffy eq $\frac{dy}{dx} = ay-by^3$, where a and b are constants? I thought about applying chain rule $\frac{dy}{dx} ...
1
vote
1answer
20 views

Stuck with weird results when solving $\frac{d^2y}{dt^2} + 4 \frac{dy}{dt} +4y = e^{-\alpha t}$.

Find the general solution of the differential equation $$\frac{d^2y}{dt^2} + 4 \frac{dy}{dt} +4y = e^{-\alpha t}$$ where $α$ is a constant and $α ≠ 2$ Normally questions which ask to ...
1
vote
0answers
14 views

Scheme for mixed partial derivative from Taylor

I need to deduce a scheme(finite difference) for the partial derivative: $$\frac{\partial^3 u}{\partial t \partial x \partial x} $$ How can I deduct it from Taylor polynomial? Thanks for your help
0
votes
1answer
45 views

Discretization of a heat equation using finite-difference method

Let $\overline{\Omega}=\left([0,2]\times [0,1]\right)\cup \left([1,2]\times [1,2]\right)$ and $\Gamma=\Gamma_1\cup\ldots\cup\Gamma_6$ be defined as shown in the following picture: I want to ...
3
votes
3answers
78 views

1-dimentional stochastic differential equation

I would like to solve this SDE $$dX_{t}=\left(\sqrt{1+X^{2}}+\dfrac{1}{2}\right)dt+\sqrt{1+X^{2}} dB_{t}$$ I've tried to solve first the homogeneous equation ...
0
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0answers
29 views

Solving initial value problem using redution of order and Green's functions

Here's the question: Consider the initial value problem (3 + t)x''(t) + (2 + t)x'(t)-x(t) = (3 + t)^2 for all t > 0 subject to x(0) = 0 and x'(0) = 1 where you are given that one independent ...
0
votes
1answer
21 views

Is this equality for derivatives true?

I was trying to solve a problem and I came across the left term in the equality below (which is part of a differential equation), were $\mu$ is a constant: ...
3
votes
1answer
20 views

Can someone explain linearisation on nonlinear systems to me?

I want to find all critical points of the following nonlinear system: $$\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}$$ $$\b y_1' \\ y_2'\e = \b 5y_2 -15 \\y_2^2 - y_1 ^2\e$$ Then use linearisation ...
0
votes
1answer
12 views

Why is the equation $\frac{(z-i)^2}{(z^2+1)^2}=\frac{1}{(z+i)^2} $ in the residue theorem accurate?

I don't understand the reasoning here: $\frac{(z-i)^2}{(z^2+1)^2}=\frac{1}{(z+i)^2} $
3
votes
2answers
49 views

How to find a conserved quantity in this differential equation.

Consider the system: $$\ddot x = x^3 -x$$ What is the method to follow to find a conserved quantity for this system? So far what I have is: $\dot x = y$ and $\dot y = x^3 - x$ and I can find the ...
0
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0answers
28 views

Maximal interval of existence and uniqueness for initial value problem [on hold]

How to determine the longest interval in which the initial value problem $$ (t-1) y'' - 3 t y' + 5 y = \sin t, \\ y(-3) = 2 y'(-3) = 1 $$ has a unique twice-differentiable solution, without solving ...
3
votes
3answers
66 views

How do we find $u(x)$?

I want to know how to find $u(x)$ in the below question: $$u''(x)+{e^u}^{(x)} = 0\\ x \in[0,1]\\u(0) = u(1) = 0$$ Please explain briefly how this was done?? Thanks!!
1
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0answers
20 views

Analytic Solution of Second Order Nonlinear odes

Any idea how to find analytic solution of the following ODE. $y''+0.1 y'+y^{5} = \sin (t)$ I will really appreciate your response! Shah
0
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3answers
49 views

PDE - Energy - Wave Equation

I dont know how solve a) and b), I'm read the book of Walter Strauss, but I have a lot of doubts, for the c) first, I tried estimate $(k(t)e^{-2at})'$ and integrate the inequal, but not unwind... :( ...
0
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0answers
34 views

RLC Circuit and 2nd order linear DE

An RLC circuit consists of a voltage source in series with a resistor, a capacitor, and an inductor. An inductor is a coil of wire. When the current passing through the coil changes, a magnetic field ...
2
votes
0answers
28 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{(k-1)A_{k-1}(t)-kA_k(t)}{\alpha+2\beta t} + \delta_{k \beta} . ...
4
votes
1answer
41 views

Find K values that make a differential equation solution stable

Given some differential equations, ie. "a", or "b": a. $$Y'''+Y''+2Y'+KY=0$$ b. $$Y'''+KY''+3KY'+2Y=0$$ How do I get the $K$ values that make the solution stable? I know that for "a", it should be ...
0
votes
2answers
27 views

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0.

$0$ is an stable equilibrium of $x' = Ax$ iff $A$ is semisimple, given that all of its eigenvalues have real part 0. I'm kind of confused here: I had understood that if all of the eigenvalues of $A$ ...
1
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0answers
61 views

Car traveling on a bumpy road (ODE)

The suspension system of a car traveling on a bumpy road has a stiffness of $k = 5\times 10^6$ N/m and the effective mass of the car on the suspension is $m = 750$ kg. The road bumps can be considered ...
0
votes
1answer
45 views

System of differential equations, need help on correcting the answer I get.

I am solving this problem: $$ x'=z-y, y'=z, z'=z-x $$. The method I used involves eigenvectors. Eigenvalues that I found are: 1, i and -i, and the solution I get is x=0, y=ce^x, z=ce^x. Everything ...
2
votes
1answer
47 views

Questions about Lyapunov functions

I'm trying to find a Lyapunov function for the nonautonomous ODE $z'=g(z)$ with $z=(x,y)$ and $$ g(x,y):=(-2x-y^4,-y-x^2). $$ For the sake of analyzing the stability of the rest point $z=0$, one does ...
0
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0answers
18 views

Explanation of the Leibniz formula

I am reading the book Solving Ordinary Differential Equations I - Nonstiff Problems (1987) by Hairer et al. My question is from Section II, chapter 2 (Order conditions for RK methods), equation 2.4. ...
2
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0answers
18 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
1
vote
1answer
17 views

how to find a function $f(x)$ such that $\lim_{x\to \infty} (\frac{y}{ f(x)})=1$

$$\frac{d^2w}{dx^2}+2\frac{dw}{dx}+5w=-5x^2-4x-2$$ Given that $$y=\frac{1}{w}$$ Where The particular solution (I found) is : $$w=e^{-x}(C \cos 2x+ E \sin 2x)-x^2$$ The general ...
1
vote
1answer
41 views

Show that $A(t)\exp(\int_{t_0}^t A(s)\,ds )=\left(\exp(\int_{t_0}^t A(s)\,ds )\right)A(t)$, when $\,A(t)\,$ is symmetric.

$A(t)$ is a symmetric matrix for $t\in [t_0,a]$. show that $$A(t)\exp\left(\int_{t_0}^t A(s)ds \right)=\exp\left(\int_{t_0}^t A(s)\,ds \right) A(t)$$ it is easy but exhausting to show for ...
0
votes
1answer
20 views

changing forms of constant of integration

In solving O.D.E in my book sometimes he changes the constant of integration in the form for example C=Sin(A) where C & A are constants obtain the general solution in an explicit form but how ...
0
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0answers
33 views

Solution of this nonlinear equation

How do I solve this equation for $y$? I can see there is a trivial solution $y=0$ but how do I get $y$ as a function of other variables? Does anyone know how to use MATLAB's fzero function to find ...
0
votes
1answer
46 views

Solve Bratu problem using Python

I am going crazy trying to solve the Bratu problem using Python: $$y''(x)+ e^{y(x)} = 0, \quad \lambda = 1, \quad x \in(0,1),$$ $$y(0) = y(1) = 0$$ I have to solve this using the tridiagonal ...
0
votes
0answers
30 views

Differential inequality involving derivatives

I'm having trouble with a differential inequality. Consider a smooth function $f(x)$ defined for $x>0$ with $f'>0$. Given $0< a < b$, show that there exists smooth functions $g(x)$ and ...
0
votes
0answers
24 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

Let $f(t), g(t)$ be polynomials, and let $y$ be a function of $t$. Given the ODE $y'' + f(t) y' + g(t) y = 0$ with initial conditions $y(0) = \alpha$ and $y'(0) = \beta$, write $y$ as the sum of a ...
0
votes
0answers
30 views

Change of Variables in Cauchy-Euler equations

So I'm working on the change of variables in the Cauchy-Euler equation. And I understand everything except one step. It's the same one step skipped in the answer to the very related question here. I ...