Tagged Questions

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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3answers
42 views

Solve the DE $t \frac{dx}{dt} = (1+2\ln(t))\tan(x) $

Ok i'm just lost $$t \frac{dx}{dt} = \left(1+2\ln(t) \right) \tan(x) $$ so ummm... $$t \, dx = \left(1+2\ln(t)\right) \tan(x) \, dt $$ then... $$ \int t \, dx = \int \left( 1+2\ln(t) ...
1
vote
2answers
29 views

Values of parameter such that $y$ satisfies DE

For what values of k does $y= 5 + 3e^{kx}$ satisfy the differential equation $$ \frac{dy}{dx} = 10 - 2y $$ Hey i normally post stuff i know about the subject but i don't really get where to start ...
1
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1answer
25 views

Does $x^*(t) =(\frac{2 - e + e^2}{2 - 2e^2})e^t + (\frac{e - 3e^2}{2 - 2e^2})e^{-t} + \frac{1}{2}te^{-t}$ contain corner points?

I want to know if $x^*(t) =(\frac{2 - e + e^2}{2 - 2e^2})e^t + (\frac{e - 3e^2}{2 - 2e^2})e^{-t} + \frac{1}{2}te^{-t}$ can contain corner points. This $x^*(t)$ is the solution to the differential ...
0
votes
1answer
19 views

Elementary Differential Equations LInear operator question

If i let $a,b,c>0$ and $L=a(d^2/dt^2)+b(d/dt)+c$. If $L[y_1]=g=L[y_2]$. How would I show that $(y1-y2)\to0$ as $t\to\infty$
0
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1answer
33 views

How to find eigenvalues and eigenfunctions of this boundary value problem?

I want to find eigenvalue and eigenfunction of this problem: $$ y''+ \lambda y=0, 0<x<l \\ y(0)=0, ly'(l)+ky(l)=0 $$ And $y'$ stands for $\frac{dy}{dx}$ and similar for $y''$. I get the ...
1
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2answers
26 views

Laplace transform of a function divided by t

Using the formula $$\mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_s^\infty F(u)~du$$ I'm trying to determine the transform with $f(t)=1-e^{-t}$. The formula gives me ...
0
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1answer
20 views

Differential equation with delay

Good evening. I have the following question about the equation with delay: How could be a correct statement for well-posedness for dalay model?
5
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0answers
102 views

General Solution of $y'(x)+p(x)e^{r(x) y(x)}=q(x)$

I solved the case for the non-homogenous constant coefficients case and I wondered if there is a way to find a general solution to a non-constant coefficient case. I don't know how to approach this at ...
0
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0answers
18 views

An Ordinary Differential Equation with time varying coefficients

Let $A$ and $B$ be complex numbers, let $\beta_1$ be real and $\beta_2=2$. Consider a following Ordinary Differential Equation: \begin{equation} \frac{ d^2 r_t}{ d t^2} + \left(\frac{A}{t^{\beta_1}} + ...
2
votes
1answer
28 views

Finding eigenfunctions and eigenvalues to Sturm-Liouville operator

I'm struggling to understand how to find the associated eigenfunctions and eigenvalues of a differential operator in Sturm-Liouville form. For instance, one question that I am trying to solve is the ...
0
votes
1answer
26 views

Proving solution behavior of a 2x2 system of ODEs with arbitrary real constant coefficients (given trace and determinant conditions)

I have a system of differential equations: x'1 = ax1 + bx2 x'2 = cx1 + dx2 where a, b, c, and d are arbitrary real numbers. I have an iff statement I'm looking to prove: Show that all solutions ...
1
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2answers
36 views

Solving $x y'(x) = \tan y(x)$

I got this differential equation, but I don't know how to find its solution. $$ x\cdot y'(x) = \tan y(x) $$ Can anyone provide me with a hint?
0
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1answer
35 views

Differentiating 8 variables

I was differentiating an equation and it led me to this: $$ \frac{PA}{1+AF}-\frac{QB}{1-BF}-\frac{XC}{1+C-CF}+\frac{YD}{1-D+DF}=0$$ I need to find F, where P+Q=1 and X+Y=1. I tried expanding ...
1
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0answers
37 views

third order linear ordinary differential equations

I have this equation: $$y'''-y' = 4Ce^x +3Ce^{-x} $$ I know that the general formula is $Y = Y_h + Y_p$ I also know that this equation can be reduced to 2nd order by integrating it so it will now ...
0
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0answers
32 views

Diagonalising Laplace--Beltrami on a Lorentzian Manifold

Is the Laplace--Beltrami operator on a Lorentzian manifold always diagonalisable?
0
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0answers
21 views

Is there a unique solution?

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a given continuous function and $t_0\in (a,b)$ a fixed point. Is it true that the following problem has a unique continuous solution ...
0
votes
1answer
16 views

First integral & uniqueness of a differential equation

Assume $x'=f(x)$ has a first integral $G(x)$. If the set $\{x\in\mathbb R: G(x)=G(x_0)\}$is bounded, why does the unique solution of $\cases{x'=f(x) & \cr x(0)=x_0}$ exist ? If $G$ is the ...
0
votes
1answer
41 views

What does $Du$ mean in a differential equation?

I'm very interested in the following work: http://maths-people.anu.edu.au/~andrews/HSU_Survey141105.pdf . Unfortunately, the author uses (in this and other papers I'm interested in) the notation $Du$. ...
1
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1answer
56 views

Dealing with Partial Differential Equations and Burger's equation

The problem is: consider Burger's equation, $$u_t +uu_x = 0 $$ $$ u(x,0) = f(x) $$ Where $$f(x) = \begin{cases} 1 - |x-2| &\mbox{if}\,\, 1\leq x \leq3, \\ 0 ...
2
votes
3answers
36 views

Differential Equation. $\frac{dy}{dx}=(x+y-1)+\frac{x+y}{log(x+y)}$

$\frac{dy}{dx}=(x+y-1)+\frac{x+y}{log(x+y)}$ The question is from IIT entrance exam practice material. I have tried substituting (x+y=t) but was stuck after some process. Please help me out with ...
0
votes
1answer
37 views

Finding the annihilator of $(2-e^x)^2$

If we expand to $e^{2x}-4e^x+4$ we get that the annihilator for the first term is $(D-2)$, for the second term it is $(D-1)$ and for the third term it is just $D$. So the annihilator for the whole ...
2
votes
1answer
36 views

Eigenvalues of an integral operator on $L^2[-1, 1]$

Find the eigenvalues of the integral operator $K: L^2[-1, 1] \to L^2[-1, 1]$ defined by $(Kx)(t) = \int_{-1}^1 (1 - 3t \tau)x(\tau) d\tau$. I began with the fact that eigenvalues must be values ...
0
votes
1answer
36 views

System of differential equations, phase portrait

Consider the system of differential equations: $$x'=-2x+y-x^3$$ $$y'=-y+x^2$$ a. Determine the fixed points. (Am I correct in thinking that to determine the fixed points, I must set x' and y'=0? I'm ...
0
votes
0answers
27 views

Solving Hermite Differential Equation through Frobenius method

I'd like to know how to solve the Hermite equation $$y''(x)−2xy'(x)+\lambda x=0 $$ using the Frobenius method. I start with the assumption that $ y = \sum a_k x^{\alpha + k}$. Then the indicial ...
0
votes
0answers
21 views

Solve $3x^4p^2-xp-y=0$

Find a general solution for the differential equation $3x^4p^2-xp-y=0$, where $p=y^{\prime}$. I know we should first seperate $y$ and do next steps. I did it and got the differential equation ...
-1
votes
2answers
50 views

Non-homogeneous Differential Equation (Stuck at integration part)

I need a hand for solving the integration part of the differential equation $y''+4y=x^2sin2x$ . $(D-2i)(D+2i)y=x^2sin2x$ , $t= \dfrac{x{^2}sin2x}{D+2i}$ $t'+2it=x^2sin2x$, $t=uv$ $v=e^{-2ix}$ ...
0
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1answer
54 views

How to prove symmetry of the following Maxwell-Bloch equations?

I have the following Maxwell-Bloch equations: $\dot{E}=-\alpha_{1} E+ k_{1}P$ $\dot{P}=-\alpha_{2}P+ k_{2}ED$ $\dot{D}=-\alpha_{3}(D-\lambda) -k_{3}EP$ In this system ...
0
votes
1answer
42 views

Help Solving coupled linear PDEs by Separation of Variables

I would like to solve the following coupled system of linear PDEs by separation of variables, where a and b are constants: ${\partial{u}\over\partial{t}} = {b-a \over a+b}u + (b+a)^2v + ...
0
votes
2answers
42 views

what is the highest order differential equation exist

As far as I know that the highest order differential equation exist is the Biharmonic equation (here). can any one correct me if there is any higher order differential equation and what is the ...
0
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0answers
18 views

If a function is continuously derivable is it also continuously differentiable?

I'm doing an computer science online test and my professor is putting loads of trick questions in in. I'm wondering whether this is also one. I have a question for which we stated at the lectures ...
0
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1answer
38 views

Eigenfunctions of a second derivative operator

Consider the operator $L :=\frac{-d^2}{dy^2}+ \alpha^2 - K(y)$ on the space of functions $f(y) $ on $H^2(-a,a) \cap H_0^1(-a,a)$. Here $K(y)$ is an even function and $\alpha >0$ is a positive real ...
0
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0answers
34 views

Solving eigenvalue problem for arbitrary first-order, linear, constant coefficient operator

I asked this question before, but I decided to expand more than just giving the question statement. I deleted the question since it had no replies; I hope that's okay. Let $\mathcal{L}$ be a linear, ...
2
votes
1answer
113 views

Solving ODE $F(t)=A(t)F'(t) $

How to solve $F(t)=A(t)F'(t) ,F(0)= I\tag 1$ All are $3 \times 3$ matrices except variable t A(t) is given and has determinant $0$. $A(t)=(I-tC_1)^{-1}t^3C_2 \tag 2$ I is a constant unit ...
0
votes
0answers
33 views

Is there any use of higher order differential equations in Computing

While I learned about differentials equation,I understood that $\frac{d}{dx}$ denotes the rate of change.Also I know that $\frac{d^2}{dx^2}$ gives the rate of $\displaystyle f'(x)$.So we usually ...
3
votes
1answer
38 views

Number of solutions of an IVP

$\dfrac{dy}{dx}=60y^{\dfrac{2}{5}}$ ,$x\gt 0$ ,$y(0)=0$ has 1.a unique solution. 2.two solutions. 3.no solution. 4.infinite number of solutions. Here f(x,y)=$60y^{\dfrac{2}{5}}$ does not satisfy ...
1
vote
1answer
24 views

Direction Field and Trajectories

I am wondering how to draw a direction field and trajectories of a system of linear equations: $$ x'= \left[ \begin{array}{ c c } 4 & -2 \\ 8 & -4 \end{array} \right] x .$$ I ...
1
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1answer
40 views

General Differential Equations Salt Tank question.

I am attempting to model a relatively easy ODE differential problem, but I seem to be missing something. The model will be distilled into a spreadsheet that uses the variables as inputs into the ...
5
votes
1answer
37 views

Recursive identity for elliptic lattice constants $\sum_{\lambda\in\Lambda\setminus0} \lambda^{-2k}$

I am stuck on Exercise 3 in these notes. To keep this question self-contained: we have $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...
1
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1answer
25 views

Converting scalar ODE to coupled system

I'm currently battling the following problem: \begin{align} u^{(iv)} (x) &= f(x)\quad\text{on }(0,1)\\ u(0) = u'(0) &= 0\\ u''(1) = u'''(1) &= 0 \end{align} which is, as I've understood, a ...
0
votes
0answers
13 views

initial value problem in differential equation

Tank A has 100 gallons of water and 50lbs of salt initially. A mixture enters the tank at a rate of 4 gal/min with a concentration of 4lbs/gal. The mixture in tank A flows into Tank B at a rate of 2 ...
0
votes
1answer
50 views

Determining when these two waves separate

There's probably something really obvious I should be getting, but I haven't yet developed the intuition for working with the wave equation. Suppose we're given the wave equation $u_{tt} = c^{2} ...
3
votes
1answer
21 views

Example of ODE $x' = f(x)$ for $f : \mathbb{R}^3 \to \mathbb{R}^3$ such that solution is bounded but not periodic

On the last page of these notes: http://www.cds.caltech.edu/archive/help/uploads/wiki/files/179/lecture5Bs.pdf the author says that "for $n \geq 3$ trajectories may wander around a bounded region ...
3
votes
3answers
89 views

Numer of solutions for IVP

Consider the initial value problem $\dfrac{dy}{dx}=3y^{{2}/{3}}$ with initial condition $y(0)=0$. How many solutions are there for this IVP? 1 2 3 4 infinitely many. Clearly, ...
1
vote
1answer
35 views

Long term behavior of the solution of $y''+\omega_0^2y=\cos(\omega t)$

Please consider the differential equation $$y''+\omega_0^2y=\cos(\omega t)\;,\;\;\;\omega_0,\omega\in(0,\infty)$$ I've actually calculated the general solution ...
1
vote
2answers
49 views

Solving the Diff. Eq: $y''+9y=36x\cos(3x)$

I'm stuck on this differential equation: $$y''+9y=36x\cos(3x), \quad \text{with }y(0)=-3, y'(0)=4$$ I know the homogenous equation is: $y_H(x)=A\cos(3x)+B\sin(3x)$ Now to find the particular ...
1
vote
1answer
18 views

convergence of the derivatives

I am trying to solve the question: Let $u_n$ a sequence converging uniformly to $u$ where $u_n\in C^3(\Omega)$ for each $n$ and $\Omega$ is a subset limited of $\mathbb{R^n}$. Suppose $u_n=0$ on ...
0
votes
0answers
20 views

What would global irregularity of the Navier-Stokes Equations do?

Suppose Terrance Tao's hints at showing finite-time blowup for the true Navier-Stokes Equations prevailed, and the Navier-Stokes Problem was solved negatively (no existence and uniqueness). What good ...
5
votes
1answer
71 views

solve$\frac{xdx+ydy}{xdy-ydx}=\sqrt{\frac{a^2-x^2-y^2}{x^2+y^2}}$

solve the differential equation. $$\frac{xdx+ydy}{xdy-ydx}=\sqrt{\frac{a^2-x^2-y^2}{x^2+y^2}}$$ The question is from IIT entrance exam paper. I have tried substituting $x^2=t\ and \ y^2=u$ but was ...
1
vote
1answer
17 views

Some issues for solving differential equations using Fourier transform

Fourier transform is a powerful tool for solving differential equations. But I don't really know when the Fourier transform will give us the full general solution if it can be used. A simple example ...
0
votes
0answers
17 views

What is the link between two definitions of the Lyapunov exponent?

Consider a first order differential equation of the form \begin{equation} \frac{d x(t)}{dt} = A(t)x(t) \;\;\;\;\;\;\;\;\; x(0) = x_0 \;\;\;\;\;\;\;\;\;\;\;\;(*) \end{equation} where $x(t) \in ...