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

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1
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0answers
22 views

Best approach to matrix representation of system of nonlinear ODEs

I have this system of ODEs: $$ \frac{dS}{dt}=\pi S-\beta S Z\\ \frac{dZ}{dt}=\alpha S Z - \delta Z $$ I am trying to rewrite them in the form : $$ \pmatrix{\dot{S}\\\dot{Z}}=\mbox{diag}(S,Z) ...
0
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4answers
358 views

Linear Independence with Absolute Value Question

Are functions $t^3$ and $|t|^3$ linearly independent on $(−∞,∞)$? I'm fairly certain $t^3$ is linearly independent, as I don't see anything that would cause it to be linearly dependent. Please do ...
-4
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0answers
31 views

A Proof for Maximal Solutions [on hold]

Let $K\subset X$ be a compact set and let $x_0\in K$. Suppose that the maximal solution $x(t)$ with initial condition $x_0$ is such that $x(t)\in K$ for all $t \in T^+$ (or $T^-$). Then prove that ...
0
votes
1answer
29 views

Advection equation with source u/x

I am trying to solve following equation: $$ u_t + u_x + \frac{u}{x} = 0 $$ With initial condition: $$ u(x,0) = 0 $$ And with boundary condition given at x = 15: $$ u(15,t) = sin (wt) $$ I tried to ...
0
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1answer
8 views

Showing uniqueness of non-linear second order differential equation with initial values with some condition.

Assume $f \in C(\mathbb{R})$ and $g\in C^1(\mathbb{R})$. Show that IVP problem $$y''+f(y)y'+g(y)=0$$$$y(a)=b , y'(a)=c $$ has a unique solution. my strategy: if assume $y=x_1$ and $y'=x_2$ ...
0
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0answers
10 views

Quadratic stability linear time varying system

Consider the linear time-varying system $$ \dot{x} = A(t) x, $$ where $x \in \mathbb{R}^n$ and $A: [0,+\infty) \rightarrow \mathbb{R}^{n\times n}$ is continuous. It is known (see for instance, [1, ...
4
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1answer
84 views
+50

How to solve this differential equation: $x^2dy-y^2dx+xy^2(x-y)dy=0$

$$x^2dy-y^2dx+xy^2(x-y)dy=0$$ What I tried: $$\frac{x^2}{y^2} \frac{dy}{dx}+x(x-y)\frac{dy}{dx}=1\\$$ Let $h=-1/x, \; k=-1/y,\; dh=1/x^2 \, dx, \; dk=1/y^2 \,dy$ ...
-3
votes
0answers
41 views

Differential equation $(x^2+6xy+2y^2)dx + 2x(x+y)dy$

To be solved the differential equation: $(x^2+6xy+2y^2)dx + 2x(x+y)dy = 0$ if $y(1)=-3$. ... Not directly integrable, so I start by setting $\dfrac{y}{x} = u \iff y = ux \iff dy = udx + xdu$. So the ...
0
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0answers
17 views

What is the asymptotic behavior of these coupled ODEs?

What is the asymptotic behavior of the following coupled equations as $x\to 0$? $-y_1''(x)+\dfrac{c_1}{x^2}y_1(x)=\dfrac{d_1}{x^a}y_2(x)$, $-y_2''(x)+\dfrac{c_2}{x^2}y_2(x)=\dfrac{d_2}{x^a}y_1(x)$, ...
0
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0answers
22 views

Variation of constants for system $x' = Ax + B(t)x$

I came across this in the proof of a theorem about the stability of the solution to $x' = Ax + B(t)x$, $x \in \mathbb{R}^n$ ( Verhulst's Nonlinear ODE's, chapter 6). The proof states that such a ...
0
votes
1answer
16 views

Finding general soon for Euler equation given a trial function

Use $y=x^r$ as a trial function to find the general solution to the Euler equation: $2x^2y''+3xy'-y=0$ ; $x>0$ I have no idea how to start this, as I am only able to work with second order ...
0
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0answers
44 views

The Runga-Kutta method with a adaptive step

I have some questions about this method. I use Richardson extrapolation for select a adaptive step [Solving Ordinary Differential Equations I - Nonstiff Problems 167-168p]. What mean $\varepsilon$ ...
0
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0answers
19 views

Uniqueness of a differential equation

Let $I_o=[t_0,t_0+T]\subset\mathbb R$, where $T>0$, $f\in C^0(I_0\times\mathbb R;\mathbb R)$ and satisfying Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
0
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1answer
63 views

Functional Tranformation - Abel equation of the first kind to canonical form

I am trying to carry out the above using the method in http://www.hindawi.com/journals/ijmms/2011/387429/#sec2 but I am a little confused about the equations (2.2). The general form (2.1) is in terms ...
0
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0answers
19 views

vector space differential equations

Hi! I am working on some differential equations homework and we are up to the linear algebra part. This particular homework set on Vector space is due, but my teacher has not taught the material yet ...
2
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1answer
25 views

Classification of pde

I got stuck on the following problem: Determine the subsets of $\mathbb{R}^2$ where the pde $$u_{xx}+2xu_xu_{xy}+yu_{yy}+yu_x=1$$ is elliptic, hyperbolic and parabolic respectively. Now, at first I ...
0
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0answers
20 views

Showing a system is fully self adjoint for general unmixed boundary conditions

I have been asked to look at the following questions and I'm struggling to solve it. Let $Ly=a_2(x)y''(x)+a_1(x)y'(x)+a_0(x)y(x) , a<x<b$ such that $L^*=L$. i.e. $L$ is a self adjoint linear ...
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0answers
8 views

Simplifing a Cauchy product to find the recurrence relation when solving a differential equation using a power series solution.

I'm having trouble finding the recurrence relation of the following non linear differential equation: $y''(x)+p(x)y'(x)+y^2(x)=0$ with $y(0)=1$ and $y'(0)=0$ where ...
0
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0answers
11 views

solving three first order differential equations simultaneously with varying coefficient

I need to solve 3 first order differential equations simultaneously. I can solve this equation when [A] is constant. But in this case, as I will explain, [A] is function of z. By omitting the uz, I ...
0
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2answers
73 views

What is the general solution for $y''e^{-y} =1$? [on hold]

how can I find the general solution for an ODE $$y''e^{-y} =1?$$ Thanks.
2
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0answers
40 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
0
votes
0answers
31 views

Invariants of a nonlinear ODE

Given a nonlinear ODE and a simple constraint $x \leq c$ for some constant $c$, how can we describe the largest set (or an approximation thereof) such that if the initial value of the solution of the ...
0
votes
1answer
47 views

How does this integration make sense?

I simply don't understand how integration can lead from: $ds^2 = a^2(t) \frac{dr^2}{1 - kr^2}$ to $s(r) = \frac{\sin^{-1}(\sqrt{k}r)}{\sqrt{k}}$ I appologize, I've never been quite capable of ...
0
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0answers
11 views

How could one go about constructing this relatively simple contagious diffusion-reaction model?

How could one go about constructing a contagious diffusion-reaction model showing the relationship between disease (e.g. Ebola) and number of available healthcare workers in an unevenly distributed ...
1
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0answers
14 views

Is there a relationship between the integrand in Green's Theorem and the test for finding an integrating factor for a differential form?

Green's Theorem has the formula $$ \int_C Mdx+Ndy=\int\int_D\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)dxdy $$ There is also a well known test for finding an integrating ...
1
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0answers
42 views

How to solve a system of two differential equations describing the concentration in a leaky tank?

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
3
votes
1answer
280 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
1
vote
3answers
31 views

The limit of a solution of the logistic equation as time tends to infinity

$$ \frac{dP}{dt} = 3P(4 - P),\quad P(0) = 2.$$ What value does $P$ approach as $t$ gets large, ie. as $t \to\infty$. How do I solve this? Is the idea to this question to first rearrange the equation ...
0
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0answers
26 views

Showing a second order DE has characteristic equation

Verify that $y''-2py'+p^2y=0$ has characteristic equation $(m-p)^2=0$ and has solution $y=e^{px}$ I began by trying to solve $r^2-2p+p^2=0$ but I'm kind of stuck where to go. Any help would be ...
1
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2answers
17 views

What is the correct answer to this diffferential equation?

[Question] When solving the differential equation: $$\frac{\mathrm dy}{\mathrm dx} = \sqrt{(y+1)}$$ I've found two ways to express $y(x)$: implicitly: $2\sqrt{(y + 1)} = x + C$ or directly: $y = ...
0
votes
0answers
30 views

How do I solve this calculs problem [closed]

a) Find the general solution of $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 0.$$ b) Solve $$\frac{d^2y}{dt^2} + 3\frac{dy}{dt} - 4y = 8\cos 2t + 6\sin 2t.$$ with $y(0) = 4$, $y'(0) = 0 $ How ...
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0answers
25 views

I still could not figure out

IT is our homework problem but I have already submit it. Today, I asked professor, but I still could follow what he said clearly. $\frac{dX}{dt} = \mu(x)$ and $X(0;x) = x$, where $x,X\in R^n$ For ...
0
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0answers
21 views

Differential equation Worded Problem [duplicate]

While filling up a chemicals container at a constant rate of 300 litres/min, the crew of a naval ship discover two leakages at the bottom of the container. They discover that the chemical is leaking ...
1
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0answers
16 views

Second Order Differential Equations - Undetermined Coefficients

When solving for this one: $y''-3y'-4y=e^{-x}$ For the trial function, let: $y=Ae^{-x}$ $y'=-Ae^{-x}$ $y''=Ae^{-x}$ $=> Ae^{-x}-3(-Ae^{-x})-4(Ae^{-x})=e^{-x}$ $=> ...
0
votes
0answers
28 views

Weight function in orthogonal polynomials

My problem is how we can obtain the weight function in orthogonal polynomials? As we all know, orthogonal polynomials are defined through below equation. That is the integral:$$\int_a^bf(x)g(x)w(x)\ ...
0
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0answers
15 views

Existence and uniqueness of SDEs depending on the expected value?

I was thinking of general mean-field SDEs. But let us just look at something really simple: $$dX_t = dt + dB_t, \quad X_0=x$$ the solution to this SDE exists in a strong sense and is: $X_t = x + t ...
0
votes
0answers
27 views

Lyapunov-Schmidt reduction.

Use Lyapunov-Schmidt reduction to find an expression, or approximation, of the set of equilibria, as a function of the parameter $\lambda$, of the planar vector field ...
0
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1answer
25 views

integrating a differential equation with two derivatives

How can I solve for y(t) in terms of x(t)? Consider the following diff equation 2y'(t) + y(t) = 2x'(t) - x(t) Thanks edit: I need the solution in terms of ...
1
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3answers
47 views

2. Differential equation with initial condition

Does my work look correct? $\frac{dy}{dt}=-3(y-1)$ with $y(2)=-3$ $$\frac{dy}{dt}=-3(y-1)$$ $$\frac{1}{y-1}dy=-3dt$$ $$\int\frac{1}{y-1}dy=\int-3dt$$ $$\ln|y-1|=-3t+C$$ $$\ln|y-1|=C-3t$$ ...
2
votes
2answers
397 views

Can anyone explain why this equation using the fundamental theorem of calculus works?

\begin{align} \left| f(b)-f(a)\right|&=\left| \int_a^b \frac{df}{dx} dx\right|\\ \ \\ &\leq\left| \int_a^b \left|\frac{df}{dx}\right|\ dx\right|. \end{align} I do not ...
1
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0answers
40 views

List of eigenvalues for the Schrödinger equation

I'm writing an algorithm which computes the eigenvalues $E$ of the Schrödinger equation with potential $V(x) = x^2$, ie the harmonic oscillator. The equation is defined as follows $$ y''(x) = ...
3
votes
2answers
52 views

Finding function $f(x)$

How do we find the function(s) $f(x)$ given that $$f(x)=\int_{0}^{x} te^tf(x-t) \ \mathrm{d}t$$ My Try : I first used the property $\int_{0}^{a}g(x) \ \mathrm{d}x=\int_{0}^{a}g(a-x) \ \mathrm{d}x$ ...
0
votes
1answer
26 views

Frobenius series method

Can someone use Frobenius series method to solve this differential equation step-by-step for educational purposes? $$x^2y''+(x^2+\dfrac{5}{36})y = 0$$ Thanks in advance.
2
votes
0answers
25 views

Geodesics on a perturbed submanifold of $\mathbb{R}^m$

Let us consider $M$, a Riemannian manifold of dimension $n$, isometrically embedded in $R^m$. Let us consider a geodesic $\gamma$ on $M$. Now, let us "perturb" (in other words, change slightly the ...
0
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0answers
20 views

Conjugacy of linear systems with one zero eigenvalue

I have a question from Hirsch, Smale, and Devaney's "Differential Equations, Dynamical Systems and an Introduction to Chaos." Consider all linear systems with exactly one eigenvalue equal to 0. ...
0
votes
0answers
11 views

To get a particular solution of Poisson's equation

How can I get a particular solution of this Poisson's equation: $\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial\phi}{\partial r})+\frac{\partial^2\phi}{\partial ...
0
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0answers
28 views

Mathematics Model for measuring the evenness of a distribution

At time $t$, the distribution for a dynamical model is: $a_1(t), a_2 (t), a_3 (t),…, a_n(t)$ as the system evolves it may be expected that if the number of samples in a species is less than the ...
0
votes
1answer
35 views

Is every smooth function Lipschitz continuous?

Is every function of class $C^∞$ also (locally) Lipschitz continuous? If so, how can this be proven?
0
votes
2answers
26 views

Finding a particular solution of an inhomogeneous ODE

How can one find a particular solution of $$y'' = 120x^4 + 180 x?$$ I assumed $$Y_p= Ax^4+ Bx^3 + Cx^2 + Dx + E.$$ I am not able to find $D$ and $E$.
0
votes
0answers
22 views

Solve the Verhulst DE $y'=\frac{y}{2}(1-\frac{y}{5})$ with initial condition $y(0) = 1$ [closed]

Solve the Verhulst DE $$y'=\frac{y}{2}\left(1-\frac{y}{5}\right)$$ with initial condition $y(0) = 1$. Can anyone please help.