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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0answers
5 views

closed and bounded form

I have this problem, Let $\omega$ a closed $1$ form in $\mathbb{R^{2}}\setminus {0} $ such that $\omega$ restricted to the set $D$ is bounded with $D=\left \{ x\in\mathbb{R} such that \left | x ...
0
votes
2answers
14 views

Is this equation homogenous or inhomogeous?

I have the following differential equation in my perturbation theory notes $y'' + 2y' = -2y$ $y(0) = 0$ It says in the following section that this equation is inhomogeneous. But I thought ...
1
vote
1answer
10 views

Sum of homogeneous and inhomogeneous solutions also form a solution

For some linear differential operator, $L$, an inhomogeneous differential equation can be formed: $$ L~y(x) = F(x) \text{ with some solution } y_p (x).$$ Similarly a homogeneous equation could be ...
0
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2answers
16 views

Greens function method for Newtonian potential

this may be a silly question but, well you know when solving for the Poisson equation that gives the Newtonian potential, $\Phi$, (for a point mass, $M$, at the origin) $$\nabla^2 \Phi = 4\pi G ...
0
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1answer
10 views

Finding the max distance between two arrays

I have the solution of an ode as an array $x(t_k)$ where $k=1,...K$. I have another array which is an approximation to the solution of the ...
2
votes
0answers
15 views

Nondimensionization of a simple system.

A damped spring mass system is modelled below: $$m\frac{d^2y}{dt^2}=F_s+F_d\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space t>0$$ ...
1
vote
1answer
18 views

Equivalent of solutions of IVP

Consider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions ...
1
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0answers
10 views

How to solve a kummer equation in term of confluent hyp

How we find the solution of Kummer equation by using confluent hypergeometric function? please do help.
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0answers
22 views

Finding Convergence Function

$$S(x) = 1+x+\frac{x^2}{2} + \frac{x^3}{1 * 3} + \frac{x^4}{2 * 4} + \cdots + \cdots \frac{x^{2n+1}}{1 * 3*5 \cdots (2n+1)} + \frac{x^{2n+2}}{2*4*6\cdots (2n+2)}$$ How is it to find the convergence ...
1
vote
1answer
15 views

Asymptotic Estimate

Consider the following Sturm–Liouville problem $$u''+\lambda u=0, \ 0<x<1$$ $$u(0)-u'(0)=0, \ u(1)+u'(1)=0.$$ Obtain an asymptotic estimate for large eigenvalues. I solved the problem and ...
1
vote
0answers
24 views

differentiable curve

I´m a little stuck with this problem, I think is false but I can´t find a counter example, here is the problem Let $\omega$ a 1-form defined in $U\subset \mathbb{R^{2}}$(it can be $\mathbb{R^{n}}$, ...
0
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1answer
9 views

Using superposition to reduce a complex solution

This is a solution to under-damped harmonic oscillation: $$x = e^{-(\frac{\beta}{2})t}[cos(\gamma t) \pm i sin(\gamma t)]$$ This is the correct reduction according to wolfram (10) $$ x_1 ...
-3
votes
0answers
27 views

Laplace with heaviside step function [on hold]

solve the IVP $$y''-5y'-14y=9t+u_3(t)+4(t-1)u_1(t), \quad y(0)=0, \quad y'(0)=10$$
1
vote
0answers
22 views

Relating Differential geometry with ODEs / conformal map

Let $f:\mathbb{R}_{>0} \times (0,2\pi) \rightarrow \mathbb{R}^3$ $$f(t,\phi) := (r(t) \cos( \phi) , r(t) \sin(\phi),z(t))$$ be a surface of revolution, where we assume that $r>0$ and ...
1
vote
1answer
36 views

Solving $ f'(x) =-\log( f(x) +a ) $

Can the solution of $$ f'(x) = -\log( f(x) + a ) $$ with $f(0)=0$ and $a \in (0,1)$ be well approximated by the Lambert W function for $x>0$? It seems that morally this might be the case (by ...
1
vote
1answer
14 views

Expectation and Variance of stochastic equation

My questions is related to this question: Stochastic Differential equation, expectation and variance I.e how do you calculate the variance and expectation of $U_t = e^{-\gamma t}U_0 + \int_0^t ...
2
votes
2answers
36 views

$\frac{d^2 y}{dx^2}-2y=2\tan^3\left(x\right)$

Problem: \begin{equation} \frac{d^2 y}{dx^2}-2y=2\tan^3\left(x\right). \end{equation} using the method of undetermined coefficients or variation of parameters, with ...
1
vote
2answers
64 views

Find all values of $\alpha$ so that all solutions approach $0$ as $x \to \infty$

Given the equation $x^2y′′+\alpha xy′+4y=0$ find all values of α so that all solutions approach zero as $x \to \infty$. Anyone have advice for this question? So I get $y = c_1 ...
2
votes
3answers
36 views

Basis of a Kernel

How would i find the basis of the kernel of the differential operator below $$8y'' + 3y' + 7y$$ We know the equation was homogenous and i believe the basis is two dimensional
-3
votes
2answers
16 views

which of the following can be a differential solution for $\frac{dy}{dt}= -Cy$ [on hold]

which of the following can be a differential solution for $\frac{dy}{dt}= -Cy$ a) $y(t)=2cos(Ct)$ b) $y(t)=5e^{Ct}$ c) $y(t)=5sin(Ct) + 2cos(Ct)$ d) $y(t)=5e^{-Ct}$ e) $y(t)=4sin(Ct)$
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0answers
25 views

How to normalize a differential equation?

I have a differential equation and I am supposed to normalize it with respect to time and amplitude. I have no idea whatsoever what this means and I was unable to find something online. Any help or ...
0
votes
1answer
13 views

A differential equation question in orthogonal trajectories

3x^2 - y^2 = c My solution is; 6x - 2y.dy/dx=0 dy/dx = 3x/y . We have also a slope dy/dx= -y/3x But here, I guess there is something wrong..
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0answers
17 views

How can I find the solution of this finite element formulation?

After have used Taylor Galerkin scheme and discretize in space with Galerkin method, I have found this finite element formulation: given $A_h^{n+1}$ and $\hat{Q}_h^{n+1}$, find $\tilde{Q}_h^{n+1} \in ...
3
votes
2answers
90 views

Direct way to solve $y' = y^2$

My problem is to solve the ODE $y' = y^2$. What I would want to do is to divide by $y^2$ then integrate : it yields $y = \frac{1}{c-x}$ on an interval where $y$ does not vanish, then by continuity it ...
1
vote
1answer
36 views

How can I solve this differential equation? please help

How can I solve this differential equation $$y'= \sin(x+y) ,\ \ y(0) = -\frac{\pi}{2}, -\infty < x < \infty$$ I tried to denote $z=x+y$ And I got unfamiliar integral. Please help. Thanks
0
votes
1answer
13 views

How to get the chemical form of the the lotka-volterra ODEs

I know how to work from a chemical equation to an ODE, as described here: http://brunel.ac.uk/~cspgoop/uploads/ode_chemical_network.pdf How do I go the other way? I want to convert the ...
1
vote
1answer
37 views

I am not understanding this step

I am starting the chapter on differential equations and have this example to work through but I do not understand a few things Solve $dy=\frac{dy}{dx}=\frac{2x(y-1)}{x^2+1}$ solution: note that ...
4
votes
2answers
23 views

Applying the Fourier transform to solve an ODE.

We are learning about fourier transfrms in class and I was wondering about solving the following ODE using this method. So, I want to solve the equation $u''(x)+u(x)=0$. Now, it is clear that a ...
2
votes
0answers
19 views

Find the initial movement of a particle

A particle with mass $m$ is moving along a curve and the force exerted on it always points towards the origin, and it´s magnitude is proportional to the distance between the particle and the origin, ...
2
votes
1answer
23 views

Proving entire function is constant

Let $f(z),z^5\bar{f}(z)$ be entire functions on $\mathbb{C}$. Show that $f$ is constant. I tried using Cauchy-Riemann quations in their polar form in order to find out the derivaties are zero and ...
1
vote
0answers
17 views

Properties of a Sturm-Liouville problem

I want to show the following problem is regular. To show a Sturm-Liovulle problem is regular we need to demonstrate that $y''+\frac{b}{a}y'+\frac{1}{a}(c+\lambda)=0$ where $p(x)=e^{\int ...
-4
votes
0answers
36 views

Maths modelling!! Please Help [on hold]

Hating my life at the moment stuck on this question... PLEASE HELP!!
2
votes
0answers
31 views

Hopf bifurcation and limit cycle

I am studying bifurcation and had a system like this: $$dx/dt=ux-y-x(x^2+y^2),$$ $$dy/dt=x+uy-y(x^2+y^2).$$ I want to determine whether a Hopf bifurcation would occur. I wrote the system into polar ...
-2
votes
1answer
35 views

Laplace Transform to solve differential equation (IVP) [on hold]

how could I use Laplace Transform to solve the following differential equation: $$y''+2y'+y=0; \;\;\;\;\;y'(0)=2\;\;\;\;\;\mbox{and}\;\;\;\;\;y(1)=2$$ The solution may involve the Heaviside step ...
0
votes
0answers
11 views

Legendre Polynomials Recursion Problem

Using the recurrence equation for Legendre Polynomials: $$(k+1)P_{k+1}(x)=(2k+1)xP_k(x)-k P_{k-1}(x) \text{ , } k \in \mathbb{N}$$ Compute the Integral: $$ \int_{-1}^1xP_k(x)P_{k+1}(x)dx $$ I am ...
0
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0answers
26 views

using wolfram alpha to solve a system of nonlinear differential equations

Will Wolfram Alpha solve a system of nonlinear differential equations with initial values and graph the solutions? Essentially, I want to hand Wolfram a first-order system with variables x_1, x_2, ...
0
votes
1answer
36 views

Nonlinear Dynamics and Chemical Reactions (Ivanova Reaction System)

I have a homework problem in which I'm given an Ivanova reaction system $X+Y \longrightarrow 2Y$, $Y+Z \longrightarrow 2Z$, $Z+X \longrightarrow 2X$, and I'm asked to write the mass-action ODEs ...
0
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0answers
18 views

Solving a PDE via method of characteristics

I'm interested in solving the following PDE via the method of characteristics: $$\frac{\partial f}{\partial t} - ax\frac{\partial f}{\partial p}+ bp \frac{\partial f}{\partial x} = 0,$$ with ...
1
vote
1answer
31 views

Differential equation $x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x)$

In a proof of the series expansion of the Lambert-W-function, I need that it is the only non-zero function satisfying: $$ x\cdot f'(x)\cdot\left(f(x)+1\right)=f(x) $$ Is it true?
0
votes
1answer
36 views

Difference between power series method and Frobenius method

There is the power series method for solving ordinary differential equations: one looks for solutions of the form $\sum c_n x^n$, and derives algebraic relations between coefficients $c_n$. Then ...
2
votes
0answers
26 views

Period of a pendulum

Consider the pendulum problem $\frac{d^2x}{dt^2}+\sin(x)=0$ $\frac{dx}{dt}(0)=v_0=0$ $x(0)=x_0$ Show that the period ...
0
votes
2answers
22 views

Solving differential equation related to sales

A company finds that the sales of a particular product are declining by $12\%$ per year. That is $$\frac{dS}{dt} =−0.12S$$ If current sales are $4000 \text{ units per year}$, find how long it will ...
3
votes
0answers
30 views

Existence of Periodic Solution

I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the ...
0
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0answers
24 views

Survival models and differential equations

I have a question regarding survival models and differential equations. Is it possible to translate survival models ( in survival analysis) into differential equations? For example can we write the ...
0
votes
2answers
17 views

Solution of $y'=xy^{1/3}, y(0)=0$ equal to $0$ in $[-c,c]$ and positive for $|x|>c$.

I'm looking for a continuous function $y(x)$ which satisfies the above and trying to make it depend on $c$ so that a solution exists for any $c>0$. I read it is possible, but I can't do it... Can ...
0
votes
1answer
21 views

Multivariable integrating factor for a non-linear ODE

I'm trying to find an integrating factor of the form $u(x,y)$ for the equation $$(a\cos xy-y\sin xy)~dx+(b\cos xy-x\sin xy)~dy=0$$ using an approach suggested in the comments for this post: ...
0
votes
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
votes
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 ...