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

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3
votes
1answer
27 views

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 $\phi(t)$ ...
3
votes
2answers
32 views

Laplace Transform of a Heaviside function

Find the Laplace transform. $$g(t)= (t-1) u_1(t) - 2(t-2) u_2(t) + (t-3) u_3(t)$$ I understand that the $\mathcal{L}\{u_c(t) f(t-c)\} = e^{-cs}*F(s)$ Finding $F(s)$ is the hard part for me. My ...
3
votes
4answers
81 views

Simple differential equation( introduction but need some basic explanation)

I have a couple of questions before I dig deeper into my calculus book. First: I have learned that $\frac{d}{dx}\frac{x}{y}$=$\frac{y x'-x y'}{y^2}$ never really gotten a proper explanation for ...
4
votes
2answers
345 views

Why do the concepts of linear algebra apply to differential equations?

A lot of the stuff we do to solve differential equations are taken word for word from linear algebra. The concept of linear independence, determinant of the Wronskian used to determine independence, ...
5
votes
1answer
627 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 + ...
5
votes
3answers
104 views

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.
3
votes
1answer
28 views

Question about assumptions for Picard-Lindelöf Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelöf Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
0
votes
0answers
63 views

The Picard-Lindelöf theorem on Wikipedia

On the Wikipedia entry of Picard-Lindelöf theorem for the local existence and uniqueness of ODE's, there is a section on the optimization of the solution's interval. There is a lemma used in this ...
1
vote
2answers
54 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 ...
1
vote
2answers
67 views

Is $ \cos² y = 0 $ a solution?

I'm studying math for school. We're solving separable differential equations. One of the exercises is: $$ \frac{\Bbb d y}{\Bbb d x} = \frac{ (\cos y)^2 \tan y }{1+x²}$$ If you separate the ...
0
votes
1answer
26 views

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 : ...
0
votes
0answers
18 views
1
vote
0answers
19 views

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 ...
3
votes
1answer
35 views

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 ...
1
vote
2answers
45 views

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 ...
0
votes
3answers
33 views

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 ...
0
votes
0answers
6 views

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. ...
0
votes
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 ...
3
votes
1answer
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 ...
3
votes
1answer
35 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?
2
votes
0answers
39 views

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; $$x(0) = 0$$ ...
1
vote
0answers
12 views

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 ...
3
votes
1answer
39 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 ...
3
votes
0answers
31 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 ...
3
votes
2answers
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 ...
3
votes
0answers
65 views

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 ...
1
vote
1answer
25 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 ...
1
vote
0answers
16 views

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) ...
0
votes
0answers
26 views

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 ...
2
votes
3answers
34 views

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 ...
1
vote
1answer
62 views

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. ...
3
votes
1answer
29 views

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 ...
1
vote
0answers
36 views

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 ...
0
votes
0answers
36 views

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) ...
2
votes
3answers
88 views

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 ...
1
vote
0answers
21 views

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 ...
0
votes
0answers
12 views

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, ...
-1
votes
0answers
10 views

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
16
votes
2answers
214 views

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)\\ ...
0
votes
1answer
12 views

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 ...
2
votes
1answer
3k views

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 ...
3
votes
1answer
43 views

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 ...
1
vote
1answer
15 views

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 ...
1
vote
1answer
62 views

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 ...
0
votes
1answer
34 views

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 ...
4
votes
1answer
40 views

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 ...
6
votes
0answers
207 views

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 ...
6
votes
1answer
67 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 ...
2
votes
3answers
35 views

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 ...
0
votes
2answers
27 views

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 $y'=f(y)$, $y(a)=y_0$? ...