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

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Stability of an equilibrium

From a Center-Manifold reduction I get the following system: $$ \begin{pmatrix}\dot x \\\dot y\end{pmatrix}=\begin{pmatrix}-y(2x^2-2xy+y^2)\\x\end{pmatrix} $$ The aim is to analyze the stability of ...
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23 views

In the Glycolysys Sel'kov model, what are the meaning of “a” and “b” values?

In the Sel'kov model of glycolysis wich I put on next $u'=-u+av+u^2v\\ v'=b-av-u^2v$ wich have a limit cycle and have all sense because it is a glycolytic cicle. What are the ...
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58 views

Is it possible to bruteforce a differential equation

Is there any method to solve differential equations which involves just a number of basic functions combined into various permutations (with various factors) which are then fed into the differential ...
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56 views

Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
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22 views

Difficulty understanding Floquet multipliers wrt Mathieu equation

We have the system $$\begin{pmatrix}y\\z\end{pmatrix}' = \begin{pmatrix}0 & 1 \\ a-2\epsilon \cos t & 0 \end{pmatrix}\begin{pmatrix}y\\z\end{pmatrix}$$ and from Abel's formula we have that ...
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43 views

Does this type of differential equation have a name?

Does a differential equation of the form: $$y''(x)+\delta(x)y=Ay$$ where $\delta(x)$ is the Dirac Delta function and $A$ is a constant have a specific name?
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16 views

Step response for different definitions of step function

I was thinking about the solution of the known problem of determining the step response for the concentration leaving a CSTR tank. The differential equation is: ...
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72 views

Do I have any hope with this PIDE?

$\frac{\omega(1-\omega)}{N_1} \frac{\partial^2 f}{\partial x_1^2} + \frac{\omega(1-\omega)}{N_2} \frac{\partial^2 f}{\partial x_2^2} + \cdots + \frac{\omega(1-\omega)}{N_k} \frac{\partial^2 ...
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27 views

two point block method for solving ODE

How to solve the ordinary differential equation $$y'(t) = -1000 y(t)+ 999 e^{-t}, \hspace{10mm} 0≤t≤5.$$ $y(t)=e^{-t}$, for $t<0$. Using two point block method $$hf_{n+1}= \frac{1}{3} (hf_{n+2} ...
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Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
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Solution to system of ordinary differential equation

Given the system: $\begin{cases}x''=2y \\ y''=-2x\end{cases} $ I found the (I think) equivalent linear equation $x^{IV}+4x=0$ First question: is the equation actually equivalent to the system? ...
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30 views

Lowering the power of a linear differential equation.

$$L(x)\equiv x^{(n)}+a_1(t)x^{(n-1)}+...+a_{n-1}x'+a_n(t)=0.$$ The solutions $x_1, x_2,...,x_m (m<n)$ are given. Linearly independent. Let us find $x_{m+1},...,x_n$ It's starts off like this, I ...
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44 views

Can a differential equation be non-linear and homogeneous at the same time?

I have searched for the definition of homogeneous differential equation. I have found definitions of linear homogeneous differential equation. Can a differential equation be non-linear and homogeneous ...
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40 views

Limits of systems of differential equations.

Consider the following system of odes involving a real parameter $\epsilon$ as follows \begin{equation*} \frac{dx_i}{dt}=f_i(x) + \epsilon^2g_i(x,y) \ , \ \ \ \frac{dy_j}{dt}=h_j(y) + ...
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33 views

Decidability - Complexity

Can someone tell me where I can get some information about the following? We have linear differential equations with polynomial coefficients depending on x. $a_n(x)y^{(n)}+ \dots ...
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two variable perturbation analysis of nonlinear set of differential equations.

I have following set of equations, $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - 2 \epsilon_2 y(t) + 2 \epsilon_1 \epsilon_2 ...
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37 views

Finding a Lyapunov Function for a system involving a trigonometric function

I'm dealing with determining if $(0,0)$ is stable or not for the following system via constructing a Lyapunov function. The system is $$ \begin{cases} x'(t)=(1-x)y+x^2\sin{(x)}& \\ ...
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49 views

Preparations to finals, validation needed

I have an exam in a few days from now and I'm very nervous. I tried to tackle this one with all I got, but I'm not sure if I'm correct. If anyone can direct me, and tell me if and where I'm doing ...
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29 views

Choose Scaling for t

My question is the last part of the d) part of the exercise 1.17 in Mark Holms' Introduction to Applied Mathematics. The exercise is given below, where I have emphasized the part of it that is my ...
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25 views

Phase line and Equilibrium Points

Consider the differential equation $dy/dt=y^8+3y^6-y^2-1$. Sketch the phase line and classify the equilibrium points. Since when $y=0$, the derivative is negative and when $y>1$ the derivative ...
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28 views

Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
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Regarding continuity and the value of the function at the point of discontinuity.

Suppose while solving a boundary value problem, we have a two piece solution $f_1(x)$ and $f_2(x)$ where $f_1(x)=f(x)$ for $x < x_0$ and $f_2(x) = f(x)$ for $x>x_0$. If there is a matching ...
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Closed representation of Ladder operators in One Dimensional Second Order Homogeneous Differential Equations

(1) Has anyone published the closed representation of ladder operators for second order differential equations? More specifically the second order differential equation $$ -\partial_x^2\Psi_m(x) + ...
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Initial value problem - How can we find the coefficients $c_j$?

We have the initial value problem $$u'(t)=Au(t) \ \ , \ \ 0 \leq t \leq T \\ u(0)=u^0 \\ u \in \mathbb{R}^m$$ A is a $m \times m$ matrix The eigenvalues of $A$ are $\lambda_j$ and the corresponding ...
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29 views

All possible flat conformal metrics of dimension greater than 2

Combining List of formulas in Riemannian geometry and Conformal symmetry, is there a proof which states $$ x^\mu \to \frac{x^\mu-a^\mu x^2}{1 - 2a\cdot x + a^2 x^2} $$ represents all possible ...
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38 views

Prove using Green's theorem that the boundary value problem has at most one solution

Prove using Green's theorem that the boundary value problem $$\frac{\partial}{\partial{x}}\left ( (1+x^2)\frac{\partial{u}}{\partial{x}}\right )+\frac{\partial}{\partial{y}}\left ( ...
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34 views

Steady state response approximation of a linear differential equation using Taylor polynomial

After thinking out how to convert a non-homogeneous linear differential equation, with a polynomial input, to a homogeneous linear differential equation in general for this question I started playing ...
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Finding conditions of non existence of Periodic orbit

$$ x'=y \mbox{ and } y'=ax-by-x^2y-x^3 $$ I need non-existence of periodic orbits. Which conditions $a$ and $b$ in $\mathbb{R}$ must satisfy? First, one can see that if $a\leq 0$, then the system has ...
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Eigenvalue problem $y''+py=0$, $y(-2)=0$, $y(2)=0$

The problem states to find the non-negative solutions to the eigenvalue problem given by $y''+py=0$ where p is a parameter which may be varied. Solving this differential equation for the general ...
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What are the critical values of α where the qualitative nature of the phase portrait for the system changes?

I was given a matrix and solved for the eigenvalues and were marked as correct but I dont know how to solve for the second part of my question. Help please. $r = -1 + \dfrac{\sqrt{100 + 44 ...
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98 views

upper bound of a differential equation solution

Let $A(t)$ be a bounded singular values matrix that is function of time, and $f(t)$ an $L^\infty$ function of time. And consider the ODE $$ \dot x = A(t) x + f(t) $$ How we can describe qualitatively ...
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Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
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Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
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Continuation of differential equation

Suppose I have a differential equation $$\dot{x} = f(x)$$ which has global solution for any initial value $x(0) \in \mathcal{S}$. Is there some theorem defining conditions under which this equation ...
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Prove the $f_1, f_2$ is a basis of linear subspace of solution of differential equation

Let $p,q \in C(\mathbb{R}), L_{pq} = \{f \in C^2(\mathbb{R}):f^{(2)} + pf^{(1)} +q f = 0\} $ For each $(a,b)^T \in \mathbb{R}^2$ there is only one $f \in L_{pq}$ with $(f(0),f'(0)) = (a,b)$ 1- ...
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$V$ is $C^1$ and $V(x_0)=0$ and $ \nabla V $ is not zero $\{ x : V(x)= c \}$ is a surface with no edge around $x_0$

I am studying lyapanov second method in stablity theory of ODE. I have encountered a geometric lemma which says the following: Assume $ V:\mathbb R^n \to \mathbb R$ is a $C^1$ and $x_0 \in \mathbb ...
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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 ...
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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: ...
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38 views

Connection between possibility of non-monotonic solutions to first-order delay differential equations and 1-d discrete dynamical systems?

Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the ...
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Is there a test for tractability of nonlinear differential equations?

After lengthy attempts at tackling the problem one might say that coming up with a closed form solution for a nonlinear differential equation is not possible - that the problem is intractable. But is ...
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34 views

ordinary differential equations

I am trying to understand how the solution of this equation goes: $$\frac{y^2-1}{y}\cdot \sin(x^3)=\frac{dy}{dx}$$ with initial condition $y(0)=-0.5$ I would like to understand if the solution can ...
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Clarification of Fuchs's theorem

Here is Fuchs's theorem My professor has been saying the last couple of classes that if $p(t)$ and $q(t)$ are polynomials, then the second order differential equation converges everywhere. He hasn't ...
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What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve? In differential algebra, ...
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42 views

General solution of $ty'+2y=4t^2$

Should we left the general solution of the differential equation $t\frac{dy}{dx}+2y=4t^2$ as $t^2y=t^4+c$ instead of $y=t^2+c/(t^2)$ ($c$ is an arbitrary constant)? Does the solution $y=t^2+c/(t^2)$ ...
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55 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
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Equillibria to Differential Equations

I am wondering what the exact definition is of an equilibrium to a differential equation. It seems like the general consensus implies that a differential equation will only have an equilibrium if it ...
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41 views

Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
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18 views

For what types of differential equations is the Laplace transform most effective?

I'm reviewing for a final exam and want to make sure I know what tools to use for what situations, and was just wondering if there were situations where the Laplace transform is unusable or less ...
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35 views

Partial Differential Equations Black Scholes Problem

Part 1) Consider the Black-Scholes problem $$\frac{\partial A}{\partial t}+\frac{\sigma^2B^2}{2}\frac{\partial^2A}{\partial B^2}+rB\frac{\partial A}{\partial B}-rA=0 ...
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ODE from systems biology, can I generalize this? Have solution but not sure how to arrive at it.

Reading a systems biology book, and it describes a model with the following ODE: $$ \frac{dY}{dt} = -\gamma Y + v_1 X_1 (T - Y) + v_2 X_2 (T - Y)$$ where $Y$, $T$, $T - Y$, $X_i$, $a$ and $v_i$ are ...