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
10 views

Analyzing differential equations system stability via phase diagram

I´m having a hard time trying to understand how to analyze stability using the phase diagram method for systems, could you please guide me? My result should be just knowing if we´re in front of a ...
0
votes
1answer
27 views

Trying to solve a ODE

I've been trying to solve this ODE, but I'm noy sure if I'm doing right: Solve $(1-xy)\,dx + x(y-x)\,dy = 0$ This equation is not exact, so we calculate its integrating factor $$ \mu = ...
0
votes
0answers
18 views

Solve system of diff equations using laplace transform and evaluate x(1)

I keep getting the wrong answer, and wolphram seems to back me up. Here's the system of equations The answer I get for $x(1)$ is 10492.1... The supposedly right answer is -1426.16 Can anyone try ...
0
votes
2answers
18 views

Solving a differential equation 1st order

The equation is $$y'(t) + 2t^{-1} y(t) = x(t) $$ I keep trying an integrating factor, since it would work out nicely if I could just use $\ln(t^2)$, but it doesn't work out. Help please!
2
votes
1answer
27 views

Solve $x^2(x^2+1)y''-2x^3y'+2(x^2-1)y=0$

Find both solutions to $x^2(x^2+1)y''-2x^3y'+2(x^2-1)y=0$, given that one of the solutions is of the form $y_1 = x^n$ or $e^{ax}$. I can solve $x^2y''+Cxy'+Dy=0$ equations, but dividing by ...
0
votes
1answer
13 views

find specific solution to intial value problem with derivatives

Find a member fo the family that is a solution with conditions $y=c_1e^x+c_2e^{-x}$ domain all real numbers with $y''-y=0$, $y(0)=0$ and $y'(0)=1$ I'm unsure what to do with the derivatives?
2
votes
1answer
18 views

Differential Equation (Advertising Model)

In the sales response to advertising model given by $S'=-(a+ rA(t)/M)S+ rA(t)$, where a,M and r are constants. Assume that $S(0) = S_0$ and that advertising is constant A over a fixed time period T , ...
3
votes
1answer
19 views

Loss of stability (unphysical energy gain) for simple pendulum equation?

I am simulating a pendulum using MATLAB and noted a curious issue. When I use zero velocity and (pi - 0.1) angular position as starting conditions for my second order ODE, the solution deviates from ...
0
votes
4answers
36 views

Solution to $\frac{d^2 y}{dt^2}+y=\sec\left(t\right)$

Is there a solution to the following differential equation? \begin{equation} \frac{d^2 y}{dt^2}+y=\sec\left(t\right) \end{equation}
0
votes
1answer
21 views

Plotting the phase portrait of $\dot x = x(x-y)$ and $\dot y = y(2x-y)$

I am trying to plot the phase portrait of $\dot x = x(x-y)$ and $\dot y = y(2x-y)$ Now I have already found the fixed points of the system, (0,0). I have also found the Jacobian of (x,y) and when ...
2
votes
0answers
21 views

Black Scholes PDE

How to show that $V_1(S,t)=S\frac{\partial V(S,t)}{\partial S} $ satisfies Black-Scholes PDE given as $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + ...
2
votes
0answers
28 views

List of ODE's that can be solved by Fourier transform

I am teaching introductory level Fourier analysis and I want to give my students some basic and some not so basic examples of how to solve ordinary differential equations with the method of Fourier ...
0
votes
1answer
16 views

Is this system of differential equations homogeneous?

I learned that a system of $k$ differential equations is homogeneous if \begin{equation} x_i' = p_{i1}(t)x_i + \cdots + p_{in}(t)x_i + g_i(t), \qquad i = 1, 2, \ldots, k \end{equation} has $g_i(t) = ...
0
votes
0answers
12 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} \text{ such that }\left | ...
0
votes
1answer
19 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
13 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
votes
2answers
29 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
votes
1answer
11 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
21 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
21 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
11 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.
0
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0answers
23 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
20 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
26 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
votes
1answer
11 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
25 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
38 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 ...
2
votes
1answer
22 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
40 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
66 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
39 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
17 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) $y(t)=4sin(Ct)$
0
votes
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
19 views

A differential equation question in orthogonal trajectories

$$3x^2 - y^2 = c$$ My solution is; $$6x - 2y{dy \over dx}=0$$ $$ {dy \over dx} = {3x \over y} $$ We have also a slope ${dy \over dx}= {-y \over 3x}$ But here, I guess there is something wrong..
3
votes
2answers
93 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
38 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
38 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
24 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
27 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!!
3
votes
0answers
32 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
37 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
12 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
votes
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
37 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
votes
0answers
19 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 ...