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

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Singular Solutions of this Equation?

How would I find the singular solutions of this equation: y = $ce^{x^2}$ + $ce^{\sin x}$ (where $c$ is a constant). It should be $x^2$ if anyone gets confused by the first part of the equation. ...
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53 views

boundary conditions after change of variables

Given the nonlinear boundary value problem on $[0,1]$ $$ a_1 y'^2 - a_2y'^{5/2} - a_3y'' + y''y'^{1/2} = 0 \quad y(0) = 0, y(1) = 1 \tag 1 $$ If I change variables $s=y'^{1/2}$, then (1) becomes the ...
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Translation invariance and finite dimension imply smoothness

Let $X$ be linear subspace of $C(\mathbb R)$, the set of continuous functions on $\mathbb R$, which is closed under translations, i.e., if $f\in X$ and $h\in\mathbb R$, then $\tau_h f\in X$, where ...
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On a specific non-linear partial differential equation

Given an $n$-dimensional variable $\mathbf{x}\in\mathbb{R}^n$ and the functions $h_i: \mathbb{R}^n \rightarrow \mathbb{R}$, $i=1,\dots,l$, we would like to find a solution of the following equation: ...
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ODE: Integration Factor

Given $(4xy)dx$+$(x^2-3y)dy=0$ I conclude that an integration factor is needed. At the end of the process I get to $\ln(u(y))=-0.5\ln|y|$. Am I allowed to use $u=1/y^{0.5}$ as an integration factor, ...
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116 views

Parabolic PDEs Maximum Principle

Consider diffusion equation for t>0 and $ \boldsymbol{x} $ in a bounded doman $ D$ in $\mathbb{R}^n$, and a given scalar field $a(\boldsymbol{x}) >0$ that is uniformly bounded and continuously ...
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49 views

Power Series for Original Differential Equation

The question: $y"+x^2y'+2xy=0$ I continue to get the incorrect answer and not sure why. I changed my indices around to make x^n all throughout and that's where the trouble starts. My answer ...
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135 views

perturbation question

I'm a little stuck with a problem and I was hoping that you guys could help. Question: A projectile is fired up from the earth with an initial velocity of $v_0$ upwards. Accounting for air resistance, ...
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87 views

Differential Equation for CDF

Consider the following differential equation $$F(cx) = F(x) + x F'(x)$$ for $c>1$. Does this differential equation belong to a some well known class? Is there a way to find all the solutions ...
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66 views

Differential equation question on oscillations

For any constant $k$, show that any solution of the equation$$x^2y''-xy'+(e^x-kx^2\sin x)y=0$$has infinitely many positive zeros. From what I've learnt, I managed to convert the equation into its ...
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Stochastic input to LTI systems

Papoulis stochastics processes book question 9-6 says: Show that if $R_\nu(t_1,t_2)=q(t_1)\delta(t1-t2)$ and $\mathbf{w''}=\mathbf{v}(t)U(t)$ and $\mathbf{w}(0)=\mathbf{w'}(0)=0$ then ...
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(Sobolev space) A star domain in $\Bbb R^n$

We know the theorem: If $\Omega$ is a star domain in $\Bbb R^n$, then $C^{\infty}(\overline{\Omega})$ is dense in $W^{k,p}(\Omega)$. It means that ...
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719 views

Torricelli's Law

A water tank is made by rotating the graph of the function x=g(y) about the y-axis. The volume of the tank at height y is then given by $V(y) =\pi\int_{0}^{y}g(u)^2 \, du$ A hole is made in the ...
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144 views

Using variation of parameters, how can we assume that nether $y_1$, $y_2$ equal zero?

Here's the problem I have been given: Use the method of variation of parameters to find a general solution of the following differential equation: $$ a_1y''+a_2y'+a_3y=f(x) $$ And linearly ...
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147 views

Monodromy Groups of Differential Equations

I have heard that monodromy groups and analytic continuation can be used to construct new solutions to a differential equation from a particular solution. What references (textbook, or papers) could I ...
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378 views

Fundamental Existence and Uniqueness Theorem

This question is for anybody who has a copy of the ODE book named "Differential Equations and Dynamical Systems (3rd ed.)" by Lawrence Perko (feel free to comment if you don't have it either). In ...
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Evaluating an integral $\int \sqrt{x^{2k}+x^{-2k} - 1} dx$ (encountered in pursuit problem)

Evaluate the indefinite integral $$\int \sqrt{x^{2k}+x^{-2k}-1} dx$$ where $k\in \mathbb{R}$. Source of inspiration: The pursuit problem of fox on rabbit. Rabbit with speed $v_R$ starts from origin ...
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76 views

Existence and Uniqueness of complex ODE's

I'm wondering if there is a theorem for the existence and uniqueness of complex ODE's. If there is, would someone mind explaining the general breadth of the theorem and/or directions to an online ...
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1k views

Checking my work on a Lennard-Jones potential problem in differential equations

The Lennard-Jones potential is $$U(r) = \left[ \left( \frac{\rho}{r} \right)^{12} - \left( \frac{\rho}{r} \right)^6 \right]$$. What is the equilibrium distance? OK, so I know that the equilibrium ...
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288 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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122 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) = ...
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217 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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71 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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56 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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165 views

What Happens At An Equilibrium Point For An Autonomous First-Order Differential Equation.

Let $\frac{dx}{dt} =f(x)$ be an autonomous first-order differential equation with equilibrium point at $x_0$. a) Suppose $f'(x_0) = 0$. What can you say about the behaviour of the solution near ...
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62 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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67 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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125 views

What does 'mod' stand for in this ODE book?

I've seen in a book some: mod(a)=b What does it mean? I've seen this in a book on solving ODE by using symmetries, but I am not sure about what it means. Stephani's Differential Equations: Their ...
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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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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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Ordinary Differential Equation and graphs theory?

Is there any application of Ordinary Differential Equation in graphs theory?
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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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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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96 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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Path traced by a water drop on an ellipsoid

If we have a smooth football in the shape of an ellipsoid, and that water runs down on its sides, can we trace the path of a water drop on it? For a sphere it seems easy because the force tangential ...
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116 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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123 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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73 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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147 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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why does a fractional differential equation have a unique solution?

Why must there be a unique solution to a linear constant-coefficient fractional differential equation of order $(n,q)$ with $\lceil\frac{n}{q}\rceil$ initial conditions? (All notation is as in Miller ...
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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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137 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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138 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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55 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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292 views

Fourier Transform of one variable in a two variable function.

I have a function in two variables, that satisfies the following PDE: \begin{equation} \frac{x-x_0}{x-x_1}\Psi_{xx}+\Psi_{yy}=0 \end{equation} Initially I did use Fourier series \begin{equation*} ...
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206 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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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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58 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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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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221 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 ...