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

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1answer
13 views

Changing a heaviside function into a one line function

$$h(t) = \left\{\begin{array}{l}1,\, \pi\leq t<2\pi\\ 0,\, 0\leq t<\pi\text{ and }t\geq2\pi\end{array}\right.$$ I need to change $h(t)$ into a one line function. I believe it to be ...
1
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1answer
19 views

How to calculate Z when doing Bernoulli differential equation?

I'm just learning how to do a Bernoulli differential equation and I'm stuck at the part where you have to use Z (others call it U). For example: When (y^-3)y' + (1/2x)y^-2 = -(1/2)X² * sin²x*cosx ...
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0answers
24 views

2nd Order Nonhomogeneous with varying coefficients

Is there a way to solve or get an analytical approximation to this equation? $z''(t) + z'(t)\frac{(\omega_0 + \Delta\omega (1 - e^{-\frac{t}{\tau }}))}{Q} + z(t)(\omega_0 + \Delta\omega (1 - ...
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2answers
40 views

Fitting driven Harmonic Oscillator

I've got some datapoints of a turning disc. It is supposed to obby the following differential equation: $I\ddot{\theta}+\gamma\dot{\theta}+k\theta=\tau$, So it should have the form of a driven ...
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1answer
24 views

How to differentiate with respect to component of a vector?

Let $\vec{\alpha}=\frac{m(\vec{x})}{x^2}\vec{x}$ where $\vec{x}=(x_1,\,x_2)$. In a book I read in Eq.(3.24), it was given that $$ \frac{\partial \alpha_1}{\partial x_1}=\frac{d m}{d ...
5
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0answers
33 views

Differential equation with shifited term

I have a differential equation (Or integral equation) of the form: $$ f(x) = a e^{-x} + b \int_0^x f(cz+dx) e^{-z} dz$$ $a,b,c,d$ are constants. I am considering whether the above equation has a ...
0
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1answer
34 views

Solve differential equation

How can we solve (if a closed form expression for f(x) can be found) the following first-order linear differential equation? $$f'(x)=f(x)\cdot (\cos x+\tan x)$$ I have found that one function which ...
2
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2answers
50 views

Solving $y' + \frac{1}{2}xy + y^{2} = 0$

I am trying to solve the ODE $$y' + \frac{1}{2}xy + y^{2} = 0.$$ Mathematica gives that the answer is $$y(x) = \frac{e^{-x^2/4}}{C + 2\int_{0}^{x/2}e^{-t^{2}}\, dt}.$$ Of course, if I take this answer ...
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2answers
17 views

Quadratic equation with several variables

How does $$y^{2} - 4y -t^{2} - C = 0$$ Become $$y = 2 \pm \sqrt{t^{2} +2C + 4}$$ I know its the quadratic formula but I dont know how it got it that point The original equation is $$\frac{dy}{dt} ...
3
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2answers
41 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
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4answers
88 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 ...
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2answers
69 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 ...
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0answers
18 views

How to find the matrix associated with the differential equation? [on hold]

How to solve ordinary differential equations using matrices.
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0answers
21 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 ...
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2answers
47 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
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3answers
34 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 ...
3
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1answer
46 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 ...
0
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1answer
28 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 : ...
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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. ...
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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
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2answers
50 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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2answers
58 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 ...
2
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1answer
60 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; $$x(0) = 0$$ ...
0
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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 ...
3
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1answer
39 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
1answer
31 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
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0answers
68 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 ...
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0answers
31 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 ...
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0answers
17 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) ...
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0answers
11 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
2
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3answers
35 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 ...
3
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1answer
41 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 ...
0
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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 ...
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1answer
26 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 ...
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
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1answer
17 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 ...
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1answer
35 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
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1answer
43 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 ...
2
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3answers
37 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 ...
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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 $y'=f(y)$, $y(a)=y_0$? ...
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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 ...
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0answers
27 views

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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0answers
16 views

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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0answers
21 views

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 = ...
0
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1answer
18 views

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 ...
6
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1answer
72 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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3answers
58 views

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 ...
1
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1answer
21 views

Norm bound on exponential matrix with eigenvalue negative real part, proof

If $A$ is $n \times n$ with negative real parts of all eigenvalues, then there exists positive $K,\alpha$ such that $$\|e^{At}\| \leq Ke^{-\alpha t}$$ Furthermore, if an eigenvalue has negative part ...
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2answers
41 views

Euler Cauchy equations, change of variables

To convert an euler cauchy: $x^{2}y''+pxy'+qy=0$ equation into a linear one we perfom the substitution $x = e^z$ from which we get: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
0
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1answer
71 views

How to integrate $\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$?

How to integrate: $$\int \frac{e^x \cos x}{\tan x+\operatorname{sec}x}dx$$ I don't really have a clue? Do I need to simplify it first somehow?