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

learn more… | top users | synonyms (1)

2
votes
0answers
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 ...
2
votes
0answers
26 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 ...
2
votes
0answers
24 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 ...
2
votes
0answers
26 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 ...
2
votes
0answers
21 views

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 ...
2
votes
0answers
15 views

Finding intersections of tori/toruses

I am looking for intersections of three tori. Is this possible? If so, how? To put things in perspective: I am looking for the coordinates of point P in space, and I have a triangle on the 'ground'. ...
2
votes
0answers
18 views

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) + ...
2
votes
0answers
36 views

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 ...
2
votes
0answers
27 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 ...
2
votes
0answers
37 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 ( ...
2
votes
0answers
31 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 ...
2
votes
0answers
44 views

Differential equations, theoretical question.

$$ L(x)=x^{(n)}+a_1(t)x^{(n-1)}+\cdots +a_{n-1}(t)x'+a_n(t)x=b(t);\qquad a_1(t),a_2(t),\ldots\in C$$ $$U_j(\varphi)= \sum_{k=0}^{n-1}(M_{jk} \varphi^{k}(\alpha)-N_{jk} \varphi^{k}(\beta))= ...
2
votes
0answers
38 views

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 ...
2
votes
0answers
75 views

Differential equations, theoretical question about lowering the power of a basic differential equation.

In the text book it says we can solve:(The area of existence and uniquesness of the equation is $G= R \times R^n$)$$x^{(n)}=f(t), f \in C(R) \tag{1}$$ the following method: Integrating (1) $n$ times ...
2
votes
0answers
46 views

Need help with this proof, I don't understand it , could anyone clarify some of the details. System of linear Differential equations.

$$(*)X'=A(t)X - system$$ $$(*)PX(\alpha)+QX(\beta)=0.$$-border conditions, where P,Q constant square matrices $n \times n $. Let $Y(t)$ be the fundamental matrix for the system $(*)$ normed for$ t= ...
2
votes
0answers
40 views

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 ...
2
votes
0answers
24 views

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 ...
2
votes
0answers
89 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 ...
2
votes
0answers
43 views

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)= ...
2
votes
0answers
22 views

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 ...
2
votes
0answers
41 views

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 ...
2
votes
0answers
35 views

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- ...
2
votes
0answers
29 views

$V$ 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 ...
2
votes
0answers
30 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 ...
2
votes
0answers
82 views

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: ...
2
votes
0answers
36 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 ...
2
votes
0answers
12 views

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 ...
2
votes
0answers
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 ...
2
votes
0answers
26 views

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 ...
2
votes
0answers
156 views

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, ...
2
votes
0answers
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 $y=t^2+c/(t^2)$ ...
2
votes
0answers
53 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). ...
2
votes
0answers
18 views

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 ...
2
votes
0answers
39 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 ...
2
votes
0answers
17 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 ...
2
votes
0answers
29 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 ...
2
votes
0answers
25 views

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 ...
2
votes
0answers
57 views

Is there an elegant proof of this elementary bifurcation theory result?

Let's suppose I have a $C^1$ function $f:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, $(x,\lambda)\mapsto f(x,\lambda)$. Suppose there is a unique solution of the equation $f(x,\lambda_1)=0$, ...
2
votes
0answers
27 views

Find the points at which an IVP admits at least two solutions

Given the IVP: $$\frac{dy}{dx} = x + |\sin(y)|$$ $$y(x_0) = y_0$$ Find the points in $\mathbb R^2$ at which this IVP admits at least two solutions. Clearly, $f(x,y) = x + |\sin(y)|$ is Lipschitz ...
2
votes
0answers
95 views

Solving second order nonhomogeneous differential equation with non-constant coefficients using Laplace Transform

$ty''(t) + y'(t) -ty(t)= tf(t)$ How to solve the problem using Laplace Transform? Using Laplace transform I got $$Y(s)= C(s^2-a^2)^{-1/2} + (s^2-a^2)^{-1/2}\int (s^2-a^2)^{-1/2}F(s)\,ds$$ where ...
2
votes
0answers
74 views

How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
2
votes
0answers
161 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
2
votes
0answers
50 views

Differential Equation $(1+x^2)y'-2xy=\cot(x)$ after integrating factor

$$(1+x^2)y'-2xy=\cot(x)$$ or $$y'=\frac{2x}{1+x^2}y+\frac{\cot(x)}{1+x^2}$$ if I use an integrating factor $(e^{\int\frac{-2x}{1+x^2}dx}=\frac{1}{1+x^2})$ I get $$\frac{y}{(1+x^2)}=\int\frac{\cot ...
2
votes
0answers
37 views

Separable equation

I am looking at this first order separable differential equation, and I am stuck. Here is the equation: $\frac{du}{dt}=u$ I seperated like this: $ \frac{du}{u}=dt $ Integrated both sides and ...
2
votes
0answers
38 views

Solving differential equation with small parameter

I am trying to solve the following equation $$\frac{x}{x+1}\frac{d^{2}\left(\phi^2\right)}{dx^{2}}+\frac{2x+1}{(x+1)^{2}}\frac{d\left(\phi^2\right)}{dx}=\frac{1}{3\phi}$$ with $x\ll1$ and ...
2
votes
0answers
23 views

Is one dimesional cubic NLS is globally wellposed in $H^{s}(\mathbb R), (0<s<1)$?

We consider the one dimensional cubic nonlinear Shr\"odinger equation (NLS): $$i\partial_{t}\phi (x,t) +\Delta \phi (x,t)= \pm |\phi (x,t)|^{2} \phi(x,t), \ (x, t\in \mathbb R),$$ $$\phi (x,0) = ...
2
votes
0answers
30 views

Zeroes of a solution to a differential equation

Show that any solution to the equation $y''+xy=0$ has at least 15 zeroes on the interval $[-25,25]$. Please give me a hint.
2
votes
0answers
23 views

Is the following complex value differential equation always has a solution?

Let $a(z)$ be a fixed complex value complex variable function, not necessarily holomorphic. Consider the following differential equation $$ \frac{\overline{\partial}f}{\partial \overline{z}}+af=0. $$ ...
2
votes
0answers
21 views

Linear ordinary differential equations and their evolution operators for measurable operators

Consider the following homogeneous IVP: $$\begin{cases} \dot{u}(t)+A(t)u(t)=0 \\ u(0)=u_0 \end{cases}$$ for $u:[0,1]\to \mathbb{R}^n$ (some interval to some finite dimensional Hilbert space, let's ...
2
votes
0answers
37 views

Differential equation of the form to find

Lets $f(z)$ is some analytic function on complex plane and $y(z)$ is known analytic function on complex plane. Problem statement: find all $f(z)$ that: $$f(z) = f(z\frac{\partial}{\partial z})y(z)$$ ...