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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3answers
40 views

Analytically solving nonlinear second order ODE

I need help with providing an answer to this nonlinear ODE $a_1 + f_1(x) + f_2(x) y' - a_2\bigg((y')^2 - y''\bigg) = 0,$ where the $a_i$'s are constants and the $f_i$'s are arbitrary functions of ...
0
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0answers
13 views

Multistep method Local Truncation Error

I was doing a practice exam for a Final I have coming up and I ran into this problem, and was unsure about how to approach b). Any advice would be greatly appreciated (As the final is in 5 hours)
1
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1answer
19 views

$x^2y''+(2x^2+x)y'+(2x^2+x)y=0$ A Bessel equation

$$x^2y''+(2x^2+x)y'+(2x^2+x)y=0$$ The solution is $$e^{-x}J_o(x)+e^{-x}Y_o(x)$$ How does one approach a problem like this?
2
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0answers
26 views

Division of two series expansions

I have the two functions $u(x)$ and $v(x)$, both of which have known basis expansions $u(x) = \sum_n a_n f_n(x)$, $v(x) = \sum_n b_n f_n(x)$. I would like to calculate the function ...
7
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1answer
62 views

Using Poincaré-Bendixson to prove that there is a periodic solution

I want to use the Poincaré-Bendixson theorem to show that there exists a nontrivial (and periodic) solution to $$z'' + [\log (z^2 +4(z')^2)]z' + z = 0.$$ Therefore I substituted $u = z'$ to get $$u' ...
2
votes
0answers
34 views

First order differential equation integrating factor is $e^{\int\frac{2}{x^2-1}}$

So i got the first order ode $$(x^2-1)\frac{dy}{dx}+2xy=x$$ I divided both sides by $x^2-1$ $$\frac{dy}{dx}+\frac{2}{x^2-1}xy=\frac{x}{x^2-1}$$ in the form $y' + p(x)y = q(x)$ So that means the ...
1
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1answer
18 views

Finding equilibrium points with two equations

I am given the following two differential equations and asked to find the equilibrium points, I've looked on Pauls online notes; however, I could not find anything that was similar to my problem, any ...
1
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1answer
30 views

On a system of ODE's

Consider the following system: $$Y'(x) = \begin{bmatrix} -2 & 1 & 0 \\ 0 & -2 & 0 \\ 3 & 2 & 1 \end{bmatrix} Y(x); \ \ Y(0) = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$$ ...
0
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1answer
31 views

Fourier Transform method to solve a parabolic PDE in $\mathbb{R^n}$

Let $b\in \mathbb{R^n}$ and $c>0$. Assume $g \in C(R^n)$ has compact support and $f = f(x,t)$, $f \in C_1^2(R^n \times [0,\infty))$ has compact support. I'm trying to solve the following IBP via ...
3
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1answer
23 views

Prove that the normal to a quadratic curve passes through a specific point

I've been asked to prove that the normal to the curve $y=2x^2 - 3x^{-1/2}$ at the point $(1,-1)$ passes through the point $(12,3)$. $\frac{dy}{dx} = 4x + \frac{3}{2}x^{-3/2}$ Hence, at the point ...
0
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1answer
16 views

Integration boundaries in simple ODE

I'm reading the wikipedia article on the integration factor. In the first section, "Use in solving first order linear ODE", they describe how to solve an ODE of the form $$ y' + P(x)y = Q(x) $$ ...
2
votes
1answer
30 views

Solutions of autonomous ODEs are monotonic

Problem. Let $I,J$ be open intervals, $\,f:I\to \mathbb R$, continuous, $\,\varphi :J\to R$, continuously differentiable, with $\varphi[J]\subset I$, and $\varphi$ satisfying $$ ...
1
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1answer
18 views

Use separation of variables to find a solution $u= u(x,t)$…

So I get up to the last paragraph of the solution. I can get the bases of the solution, but beyond that, I'm really confused as to what they did. Any help would be appreciated!
1
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0answers
17 views

Solving this population (growth) DE

At every point in time, a mass $n(t)$ enters the population with age zero. Both time $t$ and age $a$ are continuous, and $F(a, t)$ denotes the mass of people with age at or below $a$ at time $t$. ...
2
votes
1answer
35 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
0
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1answer
31 views

fourth order runge-kutta method and heavyside step function.

So I'm trying to model a hydrodynamic system that introduces a sudden "jump" in the value of a function at a specific time. The system is solved with a Runge-Kutta fourth order method. I have a ...
0
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1answer
26 views

If $f\in C^0(\overline{\Omega})$ is subharmonic and $-f$ is subharmonic, too, then $f\in C^2(\Omega)$ and $\Delta f=0$

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $f\in C^0(\overline{\Omega})$ be subharmonic, i.e. for each closed ball $\overline{B}\subseteq\Omega$ it holds: $$u\in C^2(B)\text{ is ...
1
vote
1answer
25 views

Exact equations: how do I get from a potential function to a solution?

I am able to get the potential function when solving an exact differential equation, but I don't know how to get to my solutions using $$\Psi \left (x, y \right )=c$$ $$d\Psi \left (x, y \right ...
1
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2answers
35 views

Third-order autonomous ODE

What must be true of $f:R\rightarrow R$ for it to satisfy $3f'(x)^2f''(x)-3f(x)f''(x)^2+f(x)f'(x)f'''(x)=0$? Such functions would constitute the entire class of "constant quantity pass-through" demand ...
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1answer
35 views

If $f\in C^0(\mathbb{R}^n)$ is subharmonic and $\limsup_{|x|\to\infty}f(x)\le 0$, then $f$ must be non-positive in $\mathbb{R}^n$

Let $f\in C^0(\mathbb{R}^)$ be subharmonic, i.e. for each closed ball $\overline{B}\subseteq\mathbb{R}^n$ it holds: $$u\in C^2(B)\text{ is harmonic in }B\text{ and }f\le u\text{ on }\partial ...
0
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2answers
27 views

Change of variables in differential equations

I am somewhat confused about the notation so I want to use the function variable explicit as $y(x)$ Lets say the equation is: $$x^4\frac{d^2y(x)}{dx^2}+x^3\frac{dy(x)}{dx}+y(x)=0$$ I will ...
0
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0answers
15 views

Nonlinear matrix differential equation

I am working with the following differential equation: $$\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$$ which is essentially in matrix notation: $$\dot{\mathbf{x}} = A\mathbf{x} + ...
0
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2answers
19 views

What do i let the particular solution equal in this equation $ \frac{d^2y}{dt^2}-y = e^{2t} + e^{-t}$

howdo i solve this differential equation? $$ \frac{d^2y}{dt^2}-y = e^{2t} + e^{-t}$$ so far i have that by letting $y=e^{mt}$ and plugging it in gives the complimentary solution: $$y_c(t)=C_1e^t + ...
-1
votes
2answers
31 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
22 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 ...
0
votes
1answer
19 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
36 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
15 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 ...
3
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1answer
48 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
33 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
votes
1answer
33 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
55 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
27 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
25 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 ...
1
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0answers
17 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
40 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
33 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
47 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.
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1answer
24 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
174 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
votes
1answer
16 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
15 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
votes
1answer
21 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
24 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
31 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
39 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 ...
0
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0answers
16 views

Computing Steady State Probability for 3 state markov chain

I have the equation $\frac{d}{dt}\vec{p(t)} = \vec{p(t)}Q$ here Q is a 3x3 transition matrix. $\vec{p} = (p_a,p_b,p_c)$. I have already solved Q where each row sums to 0. I have been trying to find ...
3
votes
2answers
28 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 ...
1
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2answers
32 views

First-order differential equation [closed]

What class of functions $f(x)$ satisfies $f'(x)+f(x)=k$?
1
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1answer
23 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. ...