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

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Is there a physical meaning of ranking in differential algebra?

The main stone in the Ritt's Algorithm from differential algebra is ranking. If we consider an example of a differential polynomial with two variables $x$ and $y$. Then how can we say $x$ is ranked ...
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PDE: solving Fokker-Planck equation with initial and boundary condition

Here is the problem. We have the following simple PDE: \begin{equation} \frac{\partial p(x,t)}{\partial t}= - a\frac{\partial p(x,t)}{\partial x} + \frac{D}{2} \frac{ \partial^2 p(x,t) }{\partial ...
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Differential equation: $Ay'' + By' + Cy = h(x)$

I'm stuck solving the equation $y'' - 3y' + 2y = 2x^3-30$. The auxiliary equation is $k^2 - 3k + 2 = 0$ where $k_1 = 1, k_2=3$. Thus the general solution is: $$y_g = C_1e^x + C_2e^{3x}$$ Then, I ...
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How to solve this DE?

Consider the ordinary differential equation $$y''=xyy'$$ I'm pretty stumped, so any tips on how to proceed? It seems fairly simple but I'm drawing a blank.
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Modelling interest with differential equations (Interpretation)

I am having trouble interpreting the meaning of this differential equation model for interest on an account. The problem is as follows: Assume you have a bank account that grows at an annual ...
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Determine the error constant for $y_{n+2}-4\theta y_{n+1}-(1-4\theta)y_n=h\left[(1-\theta)y_{n+2}'+(1-3\theta)y_n'\right]$

I have the following problem but I cannot solve part B in the way suggested by my professor in this past exam paper. I can solve it in a different way, but not in the specific way he's suggesting. ...
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1answer
24 views

Continuously differentiable functions are weakly differentiable

Let $\Omega\subseteq\mathbb R^n$ be a bounded domain and $u\in C^1(\Omega)$. I want to show, that $u$ is weakly differentiable, i.e. $$\int_\Omega\psi\frac{\partial u}{\partial ...
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355 views

Chebyshev Diff EQ

Find a power series solution about $x_0=0$ for the Chebyshev differential equation $$(1-x^2)y''-xy'+n^2 y=0,$$ as a function of of the integer $n$. Show that the solutions form a terminating ...
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General examples of Sturm-Liouville operators

The topic: My question pertains to examples of Sturm-Liouville operators in the context of a technical research paper on functional determinants of differential operators : ...
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31 views

Solve the following differential equation $ u_{xx}-m^2u=\delta(x-x_0)$

Find the solution of following equation $$ u_{xx}-m^2u=\delta(x-x_0),$$ $u(0)=0=u(L),\ x\in\mathbb R^2$ Actually, I don't know how to solve. Is there someone to help?
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48 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
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Is there a numerical solution for a system of three 1st order nonlinear ODE?

How would I go about solving the following system of non-linear ODEs for $x(t), y(t), z(t)$ $$x' = y $$ $$y'=\sin(x)+z$$ $$z'=y-z$$ I have the following initial conditions; $$x(0) = 0$$ ...
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Criteria when bigger number of functions can be obtained from smaller number

It is known that $$ A_1(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_1, $$ $$ A_2(x_1, x_2) = \partial \varphi(x_1, x_2)/\partial x_2 $$ holds if and only if $$ \partial A_1/\partial ...
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1answer
36 views

Solving a SDE / Finding expectation Value

I am working on a physics problem, and have come across the following stochastic differential equation: $dX(t) = \left( \frac{8}{3} X(t) - 3 X(t)^3\right)dt + dW$. I have tried all the methods to ...
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29 views

how to solve an affine differential equation

Is there a general way to solve $y'=Ay+b$, with $y, b \in \mathbb{R}^n$, $A$ a matrix, and where $A$ and $b$ are constant? I'm tempted to make the substitution $z = y+A^{-1}b$, and then use the matrix ...
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601 views

Spring Calculation - find mass

A spring with an -kg mass and a damping constant 9 can be held stretched 2.5 meters beyond its natural length by a force of 7.5 newtons. If the spring is stretched 5 meters beyond its natural length ...
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Poincaré-Bendixson theorem, periodic solutions/periodic orbits

According to my book (Hsu: ODE), a solution $\phi(t)$ to the system $x' = f(x)$ that is bounded for all $t \geq 0$ satisfies one of: 1) $\omega(\phi)$ contains an equilibrium, or 2) either $\phi(t)$ ...
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Exam question: Are zero points justified for this answer?

I just recently had an exam and had to answer the following question: Find the solution to the initial value problem $$x'(t)=\frac{1}{x(t)}; \space x(0)=1$$ and specify the maximum interval off ...
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1answer
24 views

About the boundary conditions of the Black-Scholes-Merton PDE

I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...
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Functional equation + differential equation = way of finding solution?

Question I was wondering about the following: Let's say there is a differential equation whose solution is $f$ And $f$ also satisfies a functional equation. Can anyone construct an (non-trivial) ...
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How fast is the water level falling when the water level is 12 meters high?

Water is draining from a conical tank (with vertex down) at the rate of $2m^2/s$. The tank is 16 meters high and its top radius is 4 meters. How fast is the water level falling when the water level is ...
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Separating variables by substitution in a homogenous ODE

I am brand new to ODE's, and have been having difficulties with this practice problem. Find a 1-parameter solution to the homogenous ODE:$$2xy \, dx+(x^2+y^2) \, dy = 0$$assuming the coefficient of ...
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Lyapunov equation for stability analysis - what's the point?

Straight from Wikipedia: In the following theorem $A, P, Q \in \mathbb{R}^{n \times n}$, and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite. ...
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What does a 3D periodic solution of a differential equation look like?

The Pointcare-Bendixson Theorem implies that if a solution stays in a bounded region with no equilibrium points then it is either a periodic solution or it approaches a periodic orbit as t goes to ...
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Theorem to show trajectories of differential equations are close after small change to initial condition

Consider two solutions(or trajectories), say $x_1(t)$ and $x_2(t)$, of a system of differential equaions. That is, $$ x_1'(t)=x_2'(t)=f(x,t), t\ge0. $$ Also, $\|x_2(0)-x_1(0)\|<\epsilon$ for some ...
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Stability of equilibria of a planar polynomial ODE system using Lyapunov function

I'm trying to find equilibria using Lyapunov function, but I'm just not sure whether I'm doing in the correct way. I have the following two differential equations (fixed point at the origin) ...
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3answers
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How to interpret complex eigenvectors of the Jacobian matrix of a (linear) dynamical system?

Consider a linear ODE system of the following form: $$ \frac {dx} {dt} = Ax $$ In case $A$ has real eigenvectors, I can interpret them as the directions in which the system will move, if the initial ...
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Does an exponential bound on a Lyapunov candidate imply asymptotic stability?

If I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about ...
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Are there attractive unstable equilibria of ordinary differential equations?

I wonder why in Definition 4.2 of attractivity in this document the author demands that the equilibrium be stable. Can anyone give an example of an ODE and an equilibrium which is locally attractive, ...
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Qualitative Ordinary differential equations [on hold]

Reduce the following systems of equation to a systems of first order ODE’s: 〖( d^2 y)/〖dt〗^2 〗^+3 dz/dt+2y=0 〖( d^2 z)/〖dt〗^2 〗^+3 dy/dt+2z=0
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Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the autonomous differential system $$\left\{\ \begin{array}{lcr}x'&=&-y+f(x)\\ ...
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What is a critical point in a system of equations?

I have an assignment question based around a system of nonlinear differential equations, $$ x' = f(x, y) \\ y ' = g(x, y) $$ The first part of the question is to locate and classify all the ...
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1answer
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Solve a second order DEQ using Euler's method in MATLAB

I need to solve the equation below with Euler's method: $$y''+ \pi ye^{x/3}(2y' \sin(\pi x)+\pi y\cos (\pi x)) = \frac{y}{9}$$ for the initial conditions $y(0)=1$, $y'(0)=-1/3$ So I know I ...
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Runge Kutta stability

I am facing a problem solving a ODE with a Runge-Kutta 4th order method: The expression in order to solve is : \begin{equation} Ay^{''}+By^{'}+Cy= Cu \end{equation} \begin{equation} y =OUTPUT ...
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1answer
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Flow of time-depended vector field

Suppose $X_t$ is a time-depended vector field with flow $\phi_t$, so, $\frac{d}{dt} \phi_t = X_t(\phi_t)$. Is it true that $d \phi_t(X_t(x)) = X_t(\phi_t(x))?$ This is true when $X_t$ does not ...
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1answer
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Units of ODE solution don't match

I have to solve the differential equation: $v\,'=g-cv$. Sorry in advance for lack of latex. I will learn it soon, please let me make a question using the common programming notation for my ...
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Help with Euler Equations

This is from my textbook. Can someone give me a better explanation of what to do here? What does part (a) mean, i.e., how am I supposed to write $x = ln(t)$ in terms of $\frac{dy}{dx}$ and ...
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When can you take the limit of a parameter before solving the differential equation?

Short example: consider the differential equation \begin{align*} f'(x)=\frac{k^2}{k^2+k+1}xf(x) \end{align*} where $k$ is a parameter. Wolfram Alpha tells me that the solution to this equation is ...
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How to classify/ solve this PDE?

I am searching how to solve the PDE below but I can not seem to find a decent example online. My major did not focus much in solving PDEs so I feel very deficient. I know how to solve for the steady ...
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64 views

Flow of sum of non-commuting vector fields

Let $V,W\in\Gamma(M)$ be any two vector fields. Is there any "nice" expression for the flow of $V+W$ in terms of the flow of $V$ and the flow of $W$? It would be sufficient for me to have some sort of ...
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Forming differential equation

I'm trying to get from: $$e^{\lambda t} (\frac{dN}{dt} + \lambda N) = re^{\lambda t} $$ To: $$ \frac {d}{dt}(Ne^{\lambda t}) = re^{\lambda t} $$ However I'm not sure what procedure to use to go ...
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question on second -> first order systems [duplicate]

I have heard that it is possible to write second order IVP as first order system. What are some strategies to writing $y''=xy^2$, $y(0)=1$, $y'(0)=2$ as a first order system $y'=f(y)$, $y(a)=y_0$? ...
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Solve the equation $(m^2-m-2)x=m^2+4m+3$

Here's how I solve it I think that m is the variable (am I right?). Then $$m^2x-mx-2x-m^2-4m-3=0$$ $$m^2(x-1)-m(x+4)-(2x+3)=0$$ $$D=x^2+8x+16+4(x-1)(2x+3)$$ $$=x^2+8x+16+4(2x^2-2x+3x-3)$$ ...
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Lyapunov stability at origin with identically zero test function

At the origin, determine stability of $$x' = y \\ y' = -\tan(x)$$ If we use the test function $V(x,y) = 0.5y^2 + \int_0^x tan(s)ds$, we get $\dot{V}=x'\tan x +y'y = y\tan x -y\tan x = 0$, so the ...
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Separation of variables, Homogeneous or Exact Differential equations?

So I've just encountered these three, during exams of course they don't tell you which one is to use, if you need to use separation or homogeneous or exact. I was just wondering is there like a ...
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Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \begin{equation} \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...
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The number of characteristic curves of a pde

When a partial differential equation is elliptic, $B^2-4AC\lt 0$ and eigenvalues are complex. does there exist any characteristic curves?
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Simple related rates derivative question

Rafael is walking away from a $12$-ft-tall lantern at a constant speed. If the tip of Rafael's shadow is moving twice as fast as he walks, how tall is Rafael? I'm confused on the step where $dL/dt = ...
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Finding a power series solution for a given differential equation and identifying the function represented by the power series.

Find a power series for the solution of the differential equation $y'(t)-2y(t)=0 ,\ y(0)=5$, and then identify the function represented by the power series. (I use the following information ...