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

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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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How to properly prepare for a gradute level PDE course using Evans and Strauss ' book

For my undergrad background , I have cal 1-3, linear algebra , 1 semester of ODE, 1 semester of real analysis never have any PDE before. Thus I know this background is hardly enough to do well in a ...
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34 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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72 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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58 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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78 views

Prove that a substitution reduces the ODE to a separable equation

Show that the substitution $y=at+bx+c$ changes $x'=f(at+bx+c)$ to a separable differential equation. I did the proof. I want to check if it is correct, but appart from checking that, I have a doubt, ...
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67 views

General solution to a differential equation with an initial condition

How can I find the general solution of this DE? where $D$, $R$, and $I$ are constant parameters. $$\frac{D}{2}\frac{\ d ^2 y}{\ d x^2 \,} + {(x-RI-D)}\frac{\ d y}{\ d x} + (1+\lambda){y}{} = 0$$ ...
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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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274 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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Hypergeometric Function Differential Equation

Is there some nice obvious way to see that the hypergeometric function $$_2F_1(a,b;c:z) = \sum_{i=0}^\infty \tfrac{(a)_n(b)_n}{(c)_n}\tfrac{z^n}{n!}$$ should satisfy the differential equation ...
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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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442 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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144 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) = ...
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75 views

Numerical Method for System of ODEs

I am trying to solve a system of ordinary differential equations as follows: ...
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148 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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57 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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44 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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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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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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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{ } ...
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38 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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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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141 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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49 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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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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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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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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Properties about the Generalized Hermite-type polynomial sequence $f_n(x)=e^{-p(x)}\frac{d^n}{dx^n}e^{p(x)}$

Suppose we have the Generalized Hermite-type polynomial sequences $f_n(x)=e^{-p(x)}\dfrac{d^n}{dx^n}e^{p(x)}$ , where $p(x)$ is any polynomials of degree at least $2$ . $1.$ Do these polynomial ...
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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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63 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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137 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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400 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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80 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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97 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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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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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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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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259 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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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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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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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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120 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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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 $ ...