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

learn more… | top users | synonyms (1)

1
vote
2answers
49 views

Solving system of differential equations $\dot{x}=3x - 2y$, $\dot y = 2x - y + 15 e^t \sqrt{t}$

I am having problem with system \begin{cases} \dot{x}=3x - 2y;\\ \dot y = 2x - y + 15 e^t \sqrt{t}. \end{cases} Eigenvalues are $\lambda_1=\lambda_2=1$, the only eigenvector is $V_1 = (1,1)^T.$ I ...
0
votes
2answers
34 views

Differential equation problem. Integrating the logistic equation. [duplicate]

I would like to know how to integrate or rather solve this: $$ \frac{dP}{dt} = kP(L-P). $$ I have the solution, but I would like to know how to arrive at it. I have been told it involves separation ...
-1
votes
2answers
19 views

Create a non-linear first order differential equation which can be used using the method of separation. [on hold]

In addition, I will need to solve this equation and determine the interval of existence for the solution. Any suggestions?
2
votes
1answer
32 views

Solve a second order nonlinear equation

I have a second order nonlinear equation: $$-u''+ \frac{1}{4}(u')^2+au=x^2.$$ I am only interested in the solutions in $[0, \frac{x^2}{a}+\frac{1}{a^2}]$. One paper claims without proof that the ...
1
vote
0answers
28 views

Arnold ODE Problem

Problem 1 of Section 1.2.4 of Arnold's ODE book asks, "Can the integral curves of a smooth (continuously differentiable) equation $\frac{dx}{dt} = v(x)$ approach each other faster than exponentially ...
0
votes
0answers
11 views

detailed balance condition for coupled Langevin equation

Suppose $a$ and $m$ are real variables and they satisfy the following two coupled Langevin equations: $$ \dot{a}=F_a(a,m)+\eta_a(t);\quad\dot{m}=F_m(a,m)+\eta_m(t) $$ where $\eta_a$ and $\eta_m$ are ...
0
votes
1answer
41 views

Differential equation $(x^2y^2-1)dy+2xy^3dx=0$

$(x^2y^2-1)dy+2xy^3dx=0$ problem states that $y=t^n$ must be used. Using software it seems that there is a real solution. $$\frac{1}{3} \left(-\frac{\sqrt[3]{3 \sqrt{9 c_1^4-4 c_1^2 x^6}-9 c_1^2+2 ...
1
vote
0answers
68 views

Meaning of $dx$ [duplicate]

If I remember correctly, we use $ Δx$ for changes in $x$ and when $Δx \rightarrow 0$ then $ Δx$ takes the form of $dx$?
4
votes
0answers
25 views

Solving $y^{(n)}(t)=f(t); t>0$ with initial conditions

I will use the notation $\frac{d^n y}{dt^n} \equiv y^{(n)}$. How do I solve this ODE? $$y^{(n)}(t)=f(t); t>0;\\ y(0)=y_0, y'(0)=y_1, ..., y^{(n-1)}(0)=y_{n-1}$$ What I did: The ODE is in the ...
0
votes
0answers
9 views

$0$ is an unstable equilibrium if $f$ is Lipschitz with certain conditions

Consider the following system: $$x'=-x^3-xy^2+2x^2y^2$$ $$y'=-2y+x^2y-3x^3y$$ There are two questions: The first one is to show that $(0,0)$ is uniformly asymptotically stable. The second question ...
1
vote
1answer
18 views

intial value piecwise linear ODE; slightly wrong answer

Where am I going wrong? Solve the given initial value problem. Use a graphing utility to graph the continuous function y(x). $\frac{dy}{dx} +2xy=f(x),y(0)=2$ where $f(x)=\left\{ ...
6
votes
1answer
84 views

Second order differential equation with a variable coefficient. Show |f(x)| is bounded.

Was given this question as extra credit on an ODE exam. Didn't have time during the exam to consider it, but I have since then, and I'm stumped. $$ f''(x) + f(x) = -f'(x)x^{2015}$$ $f(x)$ is twice ...
0
votes
1answer
30 views

Solving ODE for x instead of y

Find the general solution of the ODE. Give the largest interval over which the general solution is defined. Determine any transient terms in the general solution. $y dx - 4(x+y^6)dy = 0$ This is ...
0
votes
0answers
9 views

Domain of dependence of wave equation?

Is the solution is $t=R$? Because the domain of dependence of $x=0$ is $|x-0|=t$, so compared to $|x-0|=R$. I get $t=R$. Is that correct? I am not sure if my argument is sufficient. Can anyone help ...
0
votes
1answer
19 views

Fundamental solution to the bi-harmonic operator?

I am not sure about what the hint means. If $\Delta u =\frac{1}{2 \pi}(1+\log|x|)$. Since $\log|x|$ is a fundamental solution of $\Delta u =0$. Does that mean $\frac{1}{2 \pi}(1+\log|x|)$ is a ...
2
votes
3answers
50 views

How “sharp” does a cusp have to be in order for the equation to be nondifferentiable?

From a mathematical standpoint, I understand the concept of cusps: for example, a cusp exists at the origin of $y=|x|$ because one cannot take the limit from both sides, and therefore the derivative ...
3
votes
1answer
41 views

Prove the energy is constant in a PDE?

I calculated the $$ \begin{align} \frac{dE(t)}{2\,dt} & = \int_\Omega u_tu_{tt}+DuDu_t+u^3u_t\,dx \\ & =\int_\Omega [u_t(u_{tt}-\Delta u)+u^3u_t] \, dx+\int_{\partial \Omega} u_t ...
3
votes
2answers
53 views

Why is it differential equations exist on an interval instead of a domain?

I understand a domain is the set of input elements a function is defined for (and can have breaks in it e.g. union of 2 sets) and a interval is a continuous range of real numbers. Why do we speak of ...
2
votes
1answer
46 views

Solving $\frac{dy(t)}{dt} = y(t)^2-2 y(t)+2$

How can I solve the following ODE $$ \frac{dy(t)}{dt} = y(t)^2-2 y(t)+2 $$ I'm having a tough time because the differential is in terms of $dt$. My gut instinct is to integrate both sides, but to do, ...
0
votes
0answers
7 views

Finding Floquet multipliers of a system of nonlinear differential equations

So I'm wondering how exactly one can calculate the Floquet multipliers of a system of nonlinear differential equations. In this very specific case, the system in question is $$\begin{cases} ...
3
votes
2answers
43 views

Differential Equation: $\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x$

I'm trying to solve this differential equation and believe I may have solved it using the "separable equations" method. Here's my work: $$\frac{\mathrm{d} y}{\mathrm{d} x} = xy + y\sin x = y(x + ...
4
votes
1answer
48 views

Solve $y' = \frac{1}{2}\sqrt{x} + \sqrt[3]{y}$

Please help with this $$y' = \frac{1}{2}\sqrt{x} + \sqrt[3]{y}$$ Tried making $t=\sqrt[3]{y}$. Then $3t^{2}t'_{x} = \frac{1}{2}x^\frac{1}{2} + t$. $p=t'$. And then expressed $x$ and differentiated ...
1
vote
1answer
50 views

a qualitative study of $y'=x(1+{1\over y})$

If I have as initial date $y(0)=\alpha$ for $y'=x(1+{1\over y})$ , the graph of the solution y(x) is under a parable. Can i use the comparison theorem?
3
votes
2answers
36 views

How to solve linear, second order ODE with Frobenius method with a difficult recurrence relation?

The ODE in question is: $$4xy''+2y'+y=0$$ Shifting the power series of each term so that they are all raised to the power $(n+r)$ will yield this recurrence relation: $$a_{n+1}={a_n\over ...
1
vote
1answer
37 views

solving an ODE: problem with integration

I want to solve the ODE \begin{align*} - \left(|u'|^{p-2}u'\right)' & = 1 \quad \mathrm{in}\ (-a,a)\\ u(\pm a) & = 0 \end{align*} for $1<p<\infty$ and $a>0$. I thought I could do ...
1
vote
1answer
25 views

Solving linear differential equations system

Upon trying to solve this particular system , I've encountered a few problems. $$ y'=5y+4z $$ $$z'=-4y-3z$$ After solving for eigenvalues the quadratic yielded a double root at $\lambda=1$ . But I ...
0
votes
1answer
28 views

How to determine general solution using Laplace transform?

$y′′ + a^2y = 2u(t-10)$ Here $a > 0$ and is any real number. I am confused by $a^2$ value there. Can anyone show me step by step how to get to up to $Y(s)$? It would really help me understand. ...
0
votes
1answer
14 views

Can the Simple Harmonic Oscillator D.E. be Solved Using Fourier Transform?

Can I solve the S.H.O differential equation $y'' + \omega^2y = 0$ using the Fourier Transform? I tried but couldn't get anywhere with it. I just need to know if it's not possible or whether I'm ...
0
votes
0answers
17 views

representation of Eulers's equation in biharmonic form

As we know the Euler's equation $${\rm div}{\rm div}(\frac{\nabla^2F}{\|\nabla^2F\|})=0$$ Can be written in biharmonic equation form $$\Delta^2F+ (something)=0$$ I want to know in the context of solid ...
0
votes
1answer
13 views

2nd Order Homo ODE Obtain Basis When One Solution Known [Reduction of Order]

According to my text: If we know one solution $y_1$ to $y^{"} + p(x)y' + q(x)y = 0$ then a second independent solution $y_2$ can be found if we perform a reduction of order by substituting $y(x) = ...
2
votes
0answers
31 views

Master equation of chemical reaction

I have about the construction of master equation for chemical reaction i.e. I have to construct differential equations for the probability mass function for the number of particles A, B and C. When ...
0
votes
0answers
18 views

Linear systems of differential equations

I would like to see an example of a real physical situation where one can find a set of variables evolving according to a system of linear differential equations. I wasn't able to find any such ...
2
votes
0answers
38 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 ...
0
votes
1answer
115 views

How many $f(x)$ are possible satisfying $f(x)=f'(x)$ and $f(0)=f(1)=0$.

Let $f:[0,1]\to\Bbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0,1)$ and $f(0)=f(1)=0$. Then the equation $f(x)=f'(x)$ admits how many solutions? The only solution ...
1
vote
0answers
28 views

Find a Liapunov function to show asymptotically stable

Consider the system: \begin{cases} \dfrac{dx}{dt} = y \\[12pt] \dfrac{dy}{dt} = -(1+x^{2})\,y-\sin(x) \end{cases} $(0,0)$ is a critical point of this system and I need to show that it is ...
2
votes
1answer
18 views

A more general Bessel Function

I am reading a paper where the author considers the more general Bessel equation $$x^2y'' + c_1xy' + (c_2x^{\alpha} + c_3)y =0.$$ The solutions are given, referencing some archaic text that my ...
-2
votes
0answers
13 views

How to solve this differential equation y(x)? [closed]

$\lambda^2 \frac{d^2 y}{dx^2} - y + a \frac{d J}{ d x}=0$, where $J = J(x)$, $\lambda$ and $a$ are both constants. How to find $y $? e.g $y= f(J(x), x)$.
4
votes
2answers
120 views

Finding General Solutions to 2nd Order Differential Equations, Am I on the right track?

I'm studying Differential Equations, and I'm working on figuring out how to solve using Undetermined Coefficients. We've been assured that on the test we will only have to solve equations with a 2nd ...
1
vote
2answers
18 views

Solved ODE, how did answer key rewrite solution to be in this form?

I was solving the ODE $\frac{dx}{dt} = 4(x^2+1)$ with initial condition $x(\frac{\pi}{4})=1$ I got $\tan^{-1}{x} = 4t+c$ Then I plugged in the initial value and rewrote to get ...
1
vote
2answers
39 views

How to solve a linear system in matrix form using Laplace transform?

How to solve this linear system using Laplace transform? $$\mathbf X'(t)=\left[\begin{array}{r,r,r}-3&0&2\\1&-1&0\\-2&-1&0\end{array}\right]\mathbf X(t); ~~~~~~~~\mathbf ...
0
votes
1answer
36 views

How to solve the differential equation $(x^{2}t(x)^{2n} - 1)nt(x)^{n-1}dt + 2xt(x)^{3n}dx = 0$

Solve the differential equation $(x^{2}t(x)^{2n} - 1)nt(x)^{n-1}dt + 2xt(x)^{3n}dx = 0$ I guess it should become something like $x^{2}t(x)^{2n} + ct(x)^{n} + 1 = 0$ (c is a constant) but I don't ...
1
vote
1answer
14 views

Create a first order, constant coefficient, linear differential equation with a time dependent, non-homogenous term.

I will eventually need to solve this equation using integrating factors but there are a lot of necessary parts to this equation. Anyone have any suggestions?
0
votes
1answer
44 views

fundamental matrix solution

$$ \frac{dy}{dt}=\left[\begin{array}{ccc} 5&1&1\\ 1&5&1\\ 1&1&5 \end{array}\right]$$ I need to solve this problem but my answers are still uncorrect.
0
votes
1answer
28 views

Solving Laplace's equation with separation of variables [closed]

Use the method of separation of variables to derive the solution $u(x, y)$ of Laplace’s equation $$∂^2u/∂x^2 + ∂^2u/∂y^2 = 0$$ in the rectangular domain $0<x<1$, $0<y<1$. By solving ...
0
votes
0answers
21 views

Explicit formula for the (n-2)th derivative of the Jacobi equation

The $n-2$ order derivative of the Jacobi equation is given by: $$\frac{D^n}{dt^n} V_i+\sum\limits_{l=0}^{n-2} \binom{n}{k} (\nabla_{\gamma '}^{(n-2-l)}R)(\gamma ' ,\nabla_{\gamma '}^{(l)} V_i)\gamma ...
1
vote
2answers
25 views

How to solve differential equation problem involving Dirac delta function?

$$ y''+2y'+ 10y=b\,δ\left(\, t - T\,\right)\,\qquad y\left(\, 0\,\right)=3\,,\quad y'\left(\, 0\,\right)=0 $$ Can you choose values for $b$ and $T$ ( $b$ and $T$ positive numbers) such that ...
0
votes
0answers
5 views

Confusion with order of terms

I am actually studying the paper "Derivation of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surface" http://epubs.siam.org/doi/pdf/10.1137/S003613990139828X I am confused with ...
0
votes
0answers
27 views

Why does $((exp(sds/dt)f)(t) = f(t+s)$?

Why does $((exp(sds/dt)f)(t) = f(t+s)$? I know it has to do something with the fact that ds/dt is linear, but I have no clue what I'm look at here. Any help would be appreciated!
0
votes
2answers
32 views

Solutions of differential euqations in term of definite integrals

In my textbook, the authors tried to solve the differential equation: $dy/dt+ay=g(t)$, where $a$ is a constant and $g(t)$ is a function. Why the authors of my textbook tend to leave the answer in ...
1
vote
1answer
54 views
+100

A problem on infinite domain diffusion equation

Consider the following problem $$u_t-u_{xx}=p(x,t), -\infty<x<\infty,t>0$$ $$u(x,0)=0$$ $$u\rightarrow0 \text{ as } x\rightarrow \pm \infty$$ This can be solved using many sub problems as ...