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

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43 views

Behaviour of the following function

The behaviour os a variable x over time is described by $$\frac {dx} {dt} = x^2 - x$$ where t is the variable denoting time. Suppose x is negative initially. What happens to x overtime? What would ...
1
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1answer
55 views

Poincaré lemma and conservative vector fields

Let $U$ be some contractible neighbourhood of $0\in\mathbb{R}^n$ and let $X=\sum_{i=1}^nX_i\frac{\partial}{\partial x_i}$ be a (smooth) vector field on $U$. This vector field can be thought as a ...
2
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1answer
520 views

The separation of variables in a non-homogenous equation (theory clarification)

I know "copying and paste" method from resources aren't permitted but the text is fairly long and given the amount of time I have to learn PDE (as an exchange student beside having to adapt to a ...
0
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1answer
39 views

$\frac{dy}{dx}=ye^{x}-\exp\left(e^{2x}\right)$

I want to solve $$\frac{dy}{dx}=ye^{x}-\exp\left(e^{2x}\right)$$ I've found the solution to the homogenous equation: $y = C \exp(e^x)$. Please give me a hint, how to figure out the general solution? ...
0
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1answer
35 views

using Rayleigh's Quotient to determine the interval along which lamda lies to avoid considering all possible cases of lamda

I have heard in my lectures that When solving for a PDE using the separation of variables, one checks for all possible cases of $$\text{$\lambda >$0,$\lambda $=0,$\lambda <$0}$$, but this ...
5
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1answer
123 views

Calculus of variations question with two variables

If $u(x)$ and $v(x)$ satisfy $u(0)=1$, $v(0)=-1$, $u(\pi/2) =0$, $v(π/2) =0$ on extremals of functional $$ \int_0^{\pi/2}\left[\big({\frac{du}{dx}\big)^2 +\big(\frac{dv}{dx}\big)^2 +2 \,u v ...
2
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1answer
51 views

Boundary conditions that yields no solutions to the coefficient?

I want to make sure I'm not overlooking certain steps as I've already spent an hour looking through. The heat equation is given as: $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} ...
2
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1answer
50 views

Second-order differential equation solution

What must be true of $f:R\rightarrow R$ for $f(x)f''(x)+(f'(x))^3-2(f'(x))^2=0$ to hold?
2
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1answer
110 views

Proof of Hamilton's equation from integral invariant

This is from pages 273 - 274 0f Whittaker's book of analytical dynamics. Its in the public domain. Let $q_1,q_2,\ldots,q_N$ be functions of time. And let $p_1,p_2,\ldots,p_N$ also be functions of ...
2
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2answers
36 views

Zeros of weighted sum of two Bessel functions

Just a simple and very tentative query to alleviate my seemingly futile internet digging: is there anything known on the structure of the entire function given by \begin{equation} ...
1
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0answers
30 views

2nd order homogenous ODE with non-constant (but not specified) factors

In my environment, for a given efficiency function $e(x)$, I get a distribution function $F(x)$ that is the solution to $$ F''(x) - a[e'(x)F'(x) -e'(x) F(x) -e(x)F'(x)] = 0$$ So far, I have been ...
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0answers
21 views

2nd order ODE with logistic factor

I am trying to solve $$ F''(x) + e(x)b[e(x)F'(x) - (1-e(x))F(x) ] = 0$$ with $e(x)$ being the logistic function $L/(1-\exp(-k(x-x_0))$. Neither Wolfram Alpha nor Math saga found any solution for ...
3
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1answer
110 views

General solution of continuous function-dependent ODE

Given a continuous function $f:I\subseteq\mathbb R\to \mathbb R$ and an ordinary differential equation given by: $$ y''-xf(x) y' + f(x) y = 0. $$ I'd like to solve this ODE. However, I don't know how ...
0
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1answer
42 views

What are “relaxation constants” in ode systems?

In my reading, I came across the following ODE system: $$\lambda_1 \dot x = f(x,y)$$ $$\lambda_2 \dot y = g(x,y)$$, where $\lambda_1$ and $\lambda_2$ are positive constants. Then, I saw that the ...
2
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1answer
59 views

Prove that the solution of $y'+y=\arctan(e^x), y(0)=2$ admits horizontal asymptote.

Let us consider the Cauchy problem: $$y'+y=\arctan(e^x),\ \ \ \ y(0)=2$$ Prove that the function $y(x)$ admits horizontal asymptote without solving the problem.
0
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1answer
55 views

Inhomogeneous heat equation with source term orthogonality

This is a question on the lecture notes. Basically we have the usual heat equation: $$\frac{\partial y}{\partial t}(x,t)=k^2\frac{\partial^2 y}{\partial^2 x}(x,t)+F(x,t)$$ We also have the usual ...
6
votes
3answers
289 views

Approximate solution of differential equation

My task: find approximate solution as $$y = y_0(x) + y_1(x)\lambda + y_2(x)\lambda^2 + y_3(x)\lambda^3$$ of differential equation $$y' = \sin x + \lambda e^y, y(0)=1-\lambda. \ \ \ \ (*)$$ My ...
2
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1answer
39 views

2nd order h ODE with non-constant coefficient

I have $$0 = F''(x) + p(x) F'(x) + cF(x)\\ p(x) = ab(1-x)$$ where $a$, $b$, $c$ are non-zero constants. I'm not very strong in the theories of 2nd order ODE, so I google'ed some solution methods. ...
0
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1answer
18 views

“Nullifying” an ODE

Suppose we have a two-dimensional system of ODEs, $$ \begin{array}{ccc} \dot{x} & = & f(x,y)\\ \dot{y} & = & g(x,y) \end{array} $$ What can one say about the solutions of this system, ...
0
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1answer
23 views

Difficulty understanding the method of undetermined coefficients.

I have to find the particular solution for this equation: $$y'' - 4y' + y = t*e^t + t$$ My initial thought was to use linearity and find the particular solution for both $t*e^t$ and $t$ and then ...
0
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1answer
49 views

Important ODE Solutions for Solving PDEs

Which ODEs pop up most often in the study of Partial Differential Equations such as the Heat Eq, Laplace Eq, Wave Eq, etc. At least in the homogeneous case. What are their solutions? I'm going to take ...
0
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1answer
36 views

numerical methods for ODEs

I am working on this equation: $$\frac{dx}{dt}=Ax+b$$ $$c'x=d$$ Where $x$ is a vector ,A is a constant matrix, b c are constant vectors. d is a constant number. i.e. $c_1x_1(t)+\cdots+c_nx_n(t)=d$ ...
0
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2answers
47 views

How can I solve this differential equation by substitution?

The question I'm working on asks me to solve a differential equation. The question gives a "hint" of a substitution that I should use. After working on it for over an hour with no progress, I looked ...
2
votes
2answers
43 views

Linear Ordinary Differential Equation with Nonconstant Coefficients

What would be a good method for solving these equations? $y''±kx^2y=0$ As I see, it could work with power series (At least with the minus), it wouldn't work with LaPlace. Is there a better methood ...
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2answers
36 views

First-order differential equation [closed]

What class of functions $f(x)$ satisfies $f'(x)+f(x)=k$?
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1answer
52 views

Applying the Frobenius method to $x^2 y'' - 2x y' - 10y = 0$

Here is the equation: $$x^2 y'' - 2x y' - 10y = 0 \tag{E}$$ We want to find, using the method of Frobenius, a solution in the neighbourhood of $0$, which is here a regular-singular point. ...
2
votes
1answer
33 views

How can I prove that $\phi'>0$

I have the ODE $\phi''-c\phi' + f(\phi) =0$ where $c\in\mathbb{R}$, $\phi(-\infty)=0$, $\phi(\infty)=1$ and $f$ is a smooth real valued function with $f(0)=0=f(1)$. It's true that $\phi'>0$? ...
1
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1answer
156 views

easy exponential population growth problem help?

The question is: Let $C(t)$ be the number of cougars on an island at time t years (where $t > 0$). The number of cougars is increasing at a rate directly proportional to $3500 * C(t)$. Also, $C(0) ...
3
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3answers
33 views

Equivalence of Solutions to Wave Equation

The differential equation $$\ddot x = -\omega^2 x$$ apparently has solutions of $$x = Ae^{i\omega t} + Be^{-i\omega t} \tag{1}$$ AND $$x = A\sin(\omega t) + B\cos(\omega t) \tag{2}$$ AND $$x = ...
3
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1answer
34 views

No. of linearly independent bounded solutions

Let $V$ be the set of all bounded solutions of the ODE $u''(t)-4u(t)=0$ $ where$ $t \epsilon \Bbb R$. Then $V$ is a)real vector space of dimension 2 b)real vector space of dimension 1 ...
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1answer
89 views

Inequality with differential equations solutions

I would love some help working through this problem: Let $f_1,f_2,f : [0,\infty) \to \mathbb{R}$ be three bounded, continuous and absolutely Riemann integrable functions so that $|f_1(x)|, |f_2(x)| ...
1
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2answers
89 views

Inverse Laplace transform of s/s-1

Finding the inverse laplace transform: $$L^{-1}\left\{\frac{s}{s-1}\right\}$$ I wrote: $$L^{-1}\left\{\frac{s}{s-1}\right\}=L^{-1}\left\{\frac{1}{s-1}\right\} + L^{-1}\{1\}=L^{-1}\{1\} + e^{t}$$ And ...
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0answers
20 views

Charcteristic directions

Well in my assignment I need to find the characterstic directions of a PDE. A characteristic direction is a line along which the function behaves as an ODE. The problem PDE is: $$\frac{\partial^2 ...
2
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0answers
87 views

Certain Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$

Is there a lie algebra structure $ [ \;. ] $ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following: ...
0
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1answer
47 views

Cauchy problem (1+y^2)dx + (1+x^2) = 0, y(1) = 0

Since the equation is separable, I divide it by $(1+x^2)(1+y^2) $: $$ \frac{1}{1 + x^2}dx + \frac{1}{1 + y^2}dy = 0$$ After the integration I have: $$ \arctan y = \arctan x + C => y = \tan (C ...
0
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1answer
24 views

Time to stable phase for the classic Susceptible-Infected-Susceptible epidemic model

The classic Susceptible-Infected-Susceptible epidemic model is the following: Each node is in one of the two states: Susceptible or Infected: Susceptible->Infected->Recovered. Let s and i ...
0
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1answer
51 views

Rewriting multivariate second order diffrential equation as system of first order

I hope someone can shed some light on the steps taken in between, as I have the answer and the problem, but no idea how to get there: Given the second order differential equation $$\frac{\partial^2 ...
2
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1answer
74 views

Eigenfunctions of an operator using Laguerre Polynomials

I am trying to find the eigenfunctions of the following operator: $$\mathcal{L}f=(-\gamma x+\frac{\mu}{x})f_x+\mu f_{xx}$$ I know that I must somehow use Laguerre polynomials, the solutions to the ...
1
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1answer
39 views

Differential Equation Significance

If y=mx+c is an equation that can be represented on a graph paper by a Straight Line. I was curious to know how would you represent a differential equation on Graph. Partial Differential Equation ...
0
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2answers
34 views

Confusion on Initial condition on ODE

Sorry this might sound pretty dumb but I just want to clarify something. If i know the initial condition y(0) = 0, can I say y'(0) is also 0??
0
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1answer
24 views

Obtaining a numerical method of the form $y_{n+2}-y_n=h\left(\beta_0 y_n'+\beta_1 y_{n+1}'+\beta_2 y_{n+2}'\right)$ given an identity.

I am stuck in part B of this exercise (I am giving the whole exercise though). I would really appreciate it if someone could explain to me how to solve this type of integral. Thanks a lot! Question: ...
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1answer
1k views

Find equation of tangent line using differential equation: dy/dx = x(y^1/3)

The expression $\displaystyle\frac{\mathrm{d}y}{\mathrm{d}x} = x\sqrt[3]{y}$ gives the slope at any point on the graph of the function $f(x)$ where $f(2) = 8$. a. Write the equation of the ...
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1answer
52 views

Given $\frac{du}{dt}=au+1$, $u(0)=1$ and $\frac{dv}{dt}=bv+1$, $v(0)=1$, what are $u$ and $v$?

I have the following problem which I cannot solve. I get a similar result, but nothing like the actual answer... I know this is very basic, but I cannot solve it. I would really appreciate it if ...
0
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0answers
62 views

How can I solve this differential equation with upto 12th grade math?

$f''(x)+ \frac{2}{x}\cdot f'(x)$ = $c\cdot f(x)$ Where c is a constant. I tried multiplying by $e^{\int \frac{2}{x}} = x^2$ So I get, $x^2 \cdot f''(x) +2x \cdot f'(x) = c\cdot x^2\cdot f(x)$ ...
0
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1answer
70 views

Differential equation concerning echolocation

I am in the process of trying to solve a differential equation using MATLAB. Given information is the following: $$ c(z) = 4800 - 20.2090 + \frac{17.3368z}{1000}+272.9057e^\frac{-0.7528z}{1000} $$ ...
0
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1answer
48 views

What times zero is not zero

This was a test question that I got wrong and the professor did not explain it. Given the second order equation Y''=-(Z^2)y, defined for 0<=x<=L. a.) Verify that for any constants A and B, ...
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1answer
48 views

What is the rationale behind the ability to integrate both sides of a separable differential equation?

From Paul's Online Notes: What is the rationale for the last line here? Why is it possible to do this integration? (I assume the left side is being integrated with respect to y and the left side ...
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2answers
69 views

least squares using exponential model

I'm trying to fit values from this model $$y(x)=ae^{−bx}+c$$ where a, b and c are 3 different parameters that I want to find with least squares. So using least squares I want to find the value of a, b ...
0
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1answer
306 views

Laplace transform of $t^{-3/2}$

I'm trying to find $L\{t^{-3/2}\}$. $$L\left\{\frac{1}{t}\cdot \frac{1}{t^{0.5}}\right\}=\int_s^\infty \frac{\pi^{0.5}}{x^{0.5}} dx$$ I get $\infty$. Can anyone help me solve this?
0
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1answer
40 views

Obtain a higher order of accuracy for a particular Runge Kutta method than the maximum order for general problems

I am going through a past exam paper, but there is a solution which I don't understand. Thanks a lot! The problem has 2 parts, and the part I don't understand is in part B, but it is related to part ...