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

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160 views

What does $d/dx$ actually mean?

I'm starting to learn about differential equations, and I'm having trouble mentally adjusting to working with differentials as separate quantities. (I took calculus in high school and college but I ...
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120 views

Solving $ \mathbf{y'}(t) = \omega(t) + \frac{1}2\omega(t) \times \mathbf{y}(t) $

How to solve: $$ \mathbf{y'}(t) = \omega(t) + \frac{1}2\omega(t) \times \mathbf{y}(t) $$ or equally: $$y_1′(t) = \omega_1(t) + \frac{1}2(\omega_2(t)y_3(t) - \omega_3(t)y_2(t))$$ $$y_2′(t) = ...
2
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157 views

Weak/Variational Gronwall type inequality

I came across the following weak differential inequality while looking through F.Otto's paper on $L^{1}$ contraction and uniqueness of quasilinear elliptic-parabolic equation: \begin{align*} - ...
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60 views

Differential Equation Logistic Curve

NOT A DUPLICATE - see comments below I have to find P1 where the other question does not. Also the A = some function equation is different from mine. I get this far and realize if I substitute ...
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49 views

Supersingular elliptic curves- Invariant differential exact proof question

I'm writing a minor thesis about different criteria of supersingularity and I wanted to show the following from Husemöller's Elliptic Curves [Prop. 13.3.8]: An elliptic curve $E$ in characteristic ...
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42 views

dropping a particle into a vector field, part 3

Okay, so I've been independently trying to study basic systems of differential equations as they relate to dropping a particle into a vector field. I have had two previous posts on the matter trying ...
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53 views

Why do we take 2 derivatives of the right side of a heterogeneous ODE when using the method of undetermined coefficients?

Let g(x) be the right side of a heterogeneous ODE. Why do we take 2 derivatives of g(x) when using the method of undetermined coefficients? g(x), g'(x), and g''(x) is used to guess the form of the ...
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98 views

Prove that all the solutions of (2): $\frac{dy}{dt}=A(t)y+f(t)$ are bounded in $ \left[t_0,+\infty \right )$

I have a problem: Assume that system (1): $$\dfrac{dx}{dt}=A(t)x$$ is stable, where $A(t) \in C\left [t_0,+\infty \right )$, when $t \to \infty$ and $$\begin{cases} & \mathrm{ } ...
2
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41 views

When solving PDEs is there an alternative to interpolation for out-of-grid point?

I'm numerically solving a PDE where the space domain is huge. So, I often need to interpolate to get out-of-grid points needed by the finite difference algorithm. As a result, I've a lot of numerical ...
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44 views

A system of ODEs, what existence results are there?

Let $u(t) \in \mathbb{R}^n$. Are there existence results for the ODE $$C(t)u'(t) = A(t)u(t) + f(t)$$ where $A(t), C(t) \in L^\infty(0,T;\mathbb{R}^{n\times n})$, $f(t) \in L^2(0,T;\mathbb{R}^n).$ In ...
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151 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
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51 views

Solving an eigenvalue problem on the open unit rectangle

Let $\Omega=(0,1)\times(0,1)$ and consider the boundary value problem $$\begin{cases}\Delta^2u=f\\ u(x,y)=\Delta u(x,y)=0,& x,y\in\partial\Omega \end{cases}$$ I want to solve this boundary value ...
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62 views

Ordinary Differential Equation and graphs theory?

Is there any application of Ordinary Differential Equation in graphs theory?
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34 views

eigen problem for direct scattering method

Consider the KdV equation $$u_{t}+6uu_{x}+u_{xxx}=0$$ with initial condition $$u(x,0)= \begin{cases} 1 &\text{if } x \in [-1,0] ,\\ 0 &\text {if } x \in ...
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52 views

Establishing bounds of differential equation using a maximum principle

I would like to establish that the solution of $$-\epsilon u''_\epsilon+b(x)u'_\epsilon=f(x)$$ satisfies $$||u^{(k)}_\epsilon||\leq C(1+\epsilon^{-k/2}),$$ where $b,f\in C^4(\bar\Omega)$, $b(x)\geq ...
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86 views

Exact Differential Equations of Order n?

A second order ode $Py'' + Qy' + Ry = 0$ is exact if $$(Ay' + By)' = Ay'' + (A' + B)y' + B'y = Py'' + Qy' + Ry = 0$$ How can one cast the analysis of this question in terms of exact differential ...
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107 views

solve $axy''-by'+cxy=0$ step by step

Solve $$axy''-by'+cxy=0$$ step by step I know the solution is $$y=k_1x^{u}J_{u}\left(\sqrt{\frac{c}{ a}}x\right)+k_2x^{u}Y_{u}\left(\sqrt{\frac{c}{ a}}x\right)$$ Where $k_1,k_2$ are arbitrary ...
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90 views

How can I solve this PDE using change of variables?

I am currently struggling with this PDE: $$ (xy-x)u_x-(y^2+2x^2)u_y=0 $$ with the boundary condition $$ u(0,0)=0. $$ I have tried expressing it as $$ \langle u_x,u_y\rangle \cdot \langle ...
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64 views

Solution to this Poisson equation

I am struggeling with the following PDE. Does somebody here know a solution on the whole $\mathbb{R}^2$ that goes to zero for r approaching infinity? $\Delta ...
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141 views

Where to start with this non-linear first order ode

I would like to study the following system non-linear ode system because I hope to gain some insight into the curvature of a related metric. \begin{align} (q'_1 + q'_2) &= ...
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431 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
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94 views

Lambert Omega Function

I just solved a problem and I reached a point where I could no longer simplify the equation. Being as impatient as I usually am on a Friday, I plugged my final line of derivation into WolframAlpha and ...
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106 views

Calculate half life of esters

I'm trying to calculate the level of testosterone released from different testosterone esters. Here are some graphs of testosterone levels after single injections of 250mg of each ester. Testo U ...
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50 views

Search for a candidate function with specific properties?

Given the following expression: $$ \mathcal{F(p,c,r,s)} = \frac{c^2 p^2 \left(s f'(s)-2 f(s)\right)^2}{4 f(s) \left(c^2 f(s) \left(c^2 p^2 f(s)+s^2 \left(r^2-p^2\right)\right)+\left(-r^2-1\right) ...
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143 views

Diffusion in Spherical Coordinates with mixed BC

I have been working through the book "A Guide to First-Passage Processes" and wanted to branch out on my own doing a calculation similar to what occurs in chapter 6. My basic problem comes from the ...
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76 views

Does this type of bifurcation exist?

I've been checking out numerically an ODE model of a gene circuit. Just from simulations, it appears that once a parameter passes some critical value a stable fixed point splits into three other fixed ...
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41 views

Methodology to Solve a Riccati Equation

I am new to solving ODEs and need some help. I have the following SDE: $\frac{d \eta_t}{dt} = \sigma_\mu^2 - 2 \lambda \eta_t - \sigma^{-2} \eta_t^2$ $\sigma_\mu$, $\lambda$, $\sigma$ are ...
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81 views

Dynamics of solutions close to $x(0)$ of $\dot{x}=\sqrt{x}+f(t)$ for $f(t)$ small when $t \ll 1$

I was looking at the dynamics of the real solutions close to $x(0)=0$ for the non-autonomous ODE \begin{equation} \dot{x}= \sqrt{x} +f(t) \end{equation} where $f(t)>0$ is `small' for $t \ll 1$ ...
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126 views

This matrix is an attractor?

I'm trying to find for which values of $\gamma$ the matrix A is an attractor: $$ A=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ -1 & 0 & \gamma ...
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319 views

How to use the Fredholm alternative in an ODE

I have the following ordinary differential equation $$ \frac{d^2u}{dx^2} + u = \cos x$$ A particular solution to this problem is $x\sin x$, so we can say that $$ u(x) = c_1 \cos x + c_2 \sin x + ...
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51 views

Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
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65 views

Looking for online matlab-based differential equations course/text.

I am looking for an online ODE course that would be matlab/project-oriented. A full online text/course in the spirit of this linear algebra text is preferred. I know about the following CODEE and ...
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82 views

Closed form of the solution of a nonlinear differential equation

I should solve the following problem: given a function $u(x)$, the sum of the function and its reciprocal must be equal to the integral of the function raised to $k$. Taking the derivative of the two ...
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63 views

How to solve two-level Schrödinger equation using Floquet theorem?

Consider a sinusoidal driving two-level system: $$ i \left( \begin{array}{c} \dot C_1(t) \\ \dot C_2(t) \\ \end{array} \right)=\left( \begin{array}{cc} -\frac{\omega _0}{2} & ...
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128 views

Method of undetermined coefficients for the input functions associated with the unit step

I am trying to solve a second order non-homogeneous differential equation where $x(t)$ has $u(t)$, the unit step as a part. i.e. $ x(t)= f(t)u(t) $ I know how to 'guess' the particular solution for $ ...
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74 views

How to solve a system of differential equations

When I solved one problem, I faced with the need to solve the following system of differential equations: 1) $ \ddot{x}(t)-a(t)x(t)-b(t)y(t)-c(t)=0 $ 2) $ \ddot{y}(t)-d(t)y(t)-b(t)x(t)-e(t)=0 $ ...
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485 views

How to apply Duhamel's Integral

I found one good procedure for solving the simple system of two equations with reducing on Duhamel's Integral, but I have problem to apply the same procedure on system with four equations. Let's see ...
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42 views

I have an infinite solution to an ODE even though it has only a regular singular point

I have the ODE: $\displaystyle y''(x)+\frac{y'(x)}{x+1}+y(x)=0$ I know that this has a regular singular point at $x=-1$, as $(1+x)^{-1}$ has only a first order pole, and $1$ has no pole at all, and ...
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64 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...
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68 views

Stability of limit cycle

What can be said about the stability of the limit cycle for r=1 of the equation $\dot{r}=(r^2-1)\cdot (2 r \cos(\phi) - 1), \dot{\phi}=1$. This is a problem posed in Arnol'd's book on ODEs. Does ...
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69 views

Differential Equation - $y'=5|y|^{4/5}, y(0)=0$

in the spirit of this question I ask about this one. $y'=5|y|^{4/5}, y(0)=0$ If $y> 0$ then $$y'=5|y|^{4/5}\iff y'=5^{-1}y^{4/5}\iff 5^{-1}y'y^{-4/5}=1\iff y^{1/5}=x+C\\ \iff ...
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48 views

Solving more complex diferential equations

I've come up with this implicit equation $ (y')^2(2x-2x^2+2y^2)+(y')^2=1 $ and I'd like to find the function $y(x)$ (so that it's definition doesn't contain it's derivative). Only thing I've been ...
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0answers
116 views

Show that this orbit has a zero Lyapunov exponent

I'm using J.Meiss -Differential dynamical systems, and have some trouble to understand a proof about Lyapunov exponents. We have a dynamical system $$ \dot{x} = f(x), $$ with the corresponding flow $ ...
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163 views

Hodograph transformation and implicit solution of a non-linear PDE

I am trying to understand how can one apply the Hodograph transformation to a non-linear PDE. I read that this transformation implies the representation of the solution in the implicit form . So, if I ...
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82 views

Continuity of the inverse Laplace Transform

If I know $Y(s)$, can I predict when $\mathscr{L}^{-1}[Y(s)]=y(t)$ will be continuous or continuously differentiable or even stronger conditions? For example; I'm solving an ODE with the Laplace ...
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155 views

Floquet's Theory, Hills Equation

Let us examine Hill's equation $\ddot x+Q(t)x=0$, where $Q$ is piecewise continuous and with a period $T$. Let $\mu_{1,2}$ be the multiplicators. Let $\lambda$ be the characteristic exponent. How can ...
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0answers
32 views

Property of first order differential equation

I need help with following exercise: Let $f$ be real function in $R$ of class $C^1$ and $f(r)=r$. Show that if $f'(r) \lt 1$ then no solution of the equation $x'=f(x/t)$ is tangent at $0$ to ...
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34 views

Pure differential equation whose solution is a siluroid?

I am trying to find a differential equation for the siluroid that DOES NOT contain explicitly $\theta$, $\sin\theta$, or $\cos\theta$, but only $\rho$, $\dot\rho$, $\ddot\rho$. The siluroid equation ...
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53 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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138 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...