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

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23 views

Connection between these two ODE

We have the following discrete time difference-equations: $$ F(x_t) = \Delta G(x_t) + (1-a \Delta)F(x_{t+\Delta})\\ x_{t+\Delta} = \underbrace{\Delta b(1-x_t) + \Delta x_t f(F(x_t))}_{\Delta H(x_t, ...
2
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4answers
220 views

Differential equation with integration factor

I tried to solve this differential equation: $$ydx+(2xy-e^{-2y})dy=0$$ I found $e^{2y}$ as integration factor but when affect this on equation I don't get $M_{y}$=$N_{x}$ (they are not exact)... ...
3
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0answers
10 views

Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial ...
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0answers
20 views

Picard Iteration

For part b) I understand how to get to $z_1= \begin{pmatrix} t \\ 1 \\ \end{pmatrix} \quad$and because $z= \begin{pmatrix} x \\ p \\ ...
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1answer
22 views

How to use the crank-nicolson method

I'm going over my study questions for an exam I have tomorrow in Applied Numerical Methods and I know everything except for one thing. There's a sample question about using the Crank-Nicolson method, ...
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0answers
31 views

Why use the Runge-Kutta method to solve differential equations? [on hold]

Why does one use the Runge-Kutta method to solve differential equations, calculating previous and future positions? Why not use, for example, normal equations, derivatives, or something else?
2
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1answer
40 views

The mysterious $\dot{H}^{-1}$ notation.

I have encountered the $\dot{H}^{-1}$ notation in one of the SIAM Journal on Mathematical Analysis articles. It appears to be standard (or at least not uncommon) to use this one in the field, since ...
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2answers
32 views

How can I solve the following differential equation [on hold]

How can I solve the following differential equation by power series near the point $z=1$ $$(z^2-2z+2)w''+2(z-1)w'=0$$ Then I have to find the radius of convergence of the solution
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2answers
21 views

What is my error in applying this Laplace Transform?

So, our book has the seemingly innocuous problem: $y''-y'-6y=0$. I was able to solve by hand, and come up with $${\scr L}(y)=\frac{s-2}{s^{2}-s+6}$$.That completed, I factored the bottom to ...
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1answer
24 views

Where do I begin to solve this ODE?

I've a problem: $ye^{xy}+4y^3+(xe^{xy}+12xy^2-2y)y'=0 $ I can't begin to solve it because I don't know where to start. Help me, please. Thanks!
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1answer
30 views

Write the piecewise function in terms of unit step functions

Write the piecewise function $f(t) = \begin{cases} 2t, & 0\leq t < 3 \\ 6, & 3 \le t < 5 \\ 2t, & t \ge 5 \\ \end{cases} $ in terms of unit step ...
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0answers
24 views

Is $\cosh(t^2)$ of exponential order?

Is $\cosh(t^2)$ of exponential order? I know that it isn't, but I am unsure as to why. Also why is $\cosh(t) $ of exponential order?
2
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1answer
21 views

Are the solutions of a Sturm-Liouville equation entire in the spectral parameter?

In $[1]$ the following (paraphrased) claim is made: Let $q\in L^1_{loc}([0,\infty);\mathbb{R})$, and suppose $\varphi$ and $\theta$ solve the one-dimensional Schrödinger equation \begin{equation} ...
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0answers
29 views

Derivation of the equations of APF

I am working to use the artificial potential field APF method for path planning of mobile robot; actually I found in one of references the following description about this method: the artificial ...
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0answers
23 views

1st order ODE separable

everyone! :-) I've a ODE question with I can't solve. It's here: ${dy\over dx} = {{xy + 2y-x-2}\over {xy-3y+x-3}} $ I tried the following: ${dy\over dx} = {{xy + 2y-x-2}\over ...
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1answer
55 views

$f$ satisfies $y''+by=0 \implies f$ is of class $C^{\infty}$

I know that the solutions are a linear polynomial (if $b=0$), a linear combination of exponencials ($b<0$) or linear combinations of sine and cosine ($b>0$). However, supposing I didn't know ...
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0answers
21 views

$u_{xx}+u_{yy}=0,1<r<3,0<\theta<\frac{\pi}{2}$

$u_{xx}+u_{yy}=0,1<r<3,0<\theta<\frac{\pi}{2}$ $u(1,\theta)=u(3,\theta)=0,0\leq\theta\leq\frac{\pi}{2}$ $u(r,0)=(r-1)(r-3),u(r,\frac{\pi}{2})=0,1\leq r\leq3$ I have no idea how to solve ...
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0answers
22 views

Henon (two-dimensional mapping with a Strange Attractor) [on hold]

Do you agree with Henon's statement in his paper (A two-dimensional mapping with a Strange Attractor) that one of the eigenvectors corresponding to eigenvalues calculated in equation 13 appear to be ...
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0answers
23 views

Asymptotic behaviour of ODE

I want to understand how a solution of the ODE below behaves at infinity. $$w(x)’’ = A w(x)^{-\alpha} + B w(x)^{-\beta} + C $$, with $A,B,C$ real constants and $\alpha$ and $\beta$ real, positive ...
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1answer
41 views

Uniqueness of solution

Solve the differential equation $\dot x=1+x^2$, with $x(0) = x_0$. Why can you be certain that the solution is unique? Show that for all values of $x_0$ the solution goes to infinity in finite ...
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1answer
34 views

Solving 2nd Order ODE w/Laplace Transforms

I am having difficulty with this problem: *Note: The Delta3(t) is the delta dirac function, also the answer in the image is WRONG. Attempt at solution : Let Laplace{y(t)}=Y Take Laplace of LHS ...
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0answers
15 views

A question regarding space-state representation

First of all I am not sure if this is the right place to ask this. Lets say we have a system in a form of a harmonic oscillator desribed by a second order DE. There will be 2 state variables - x ...
1
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1answer
27 views

Laplace transform nonlinear equation

How can I apply the Laplace transform on a the following nonlinear PDE $$ \frac{\partial y}{\partial t}=\frac{\partial y^n}{\partial x}$$ where $n$ is a natural number? When I say apply the Laplace ...
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1answer
325 views

Solve the initial value problem, $y'-xe^y=2e^y$, $y(0)=0$

By using an integrating factor, I get a general solution of $$y=\ln\left(\dfrac{1}{\dfrac{x^2}{2}+2x-c}\right)$$ Applying the initial condition I find $c=-1$ but I'm not sure if this is correct. To ...
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1answer
40 views

Differential Equations: solve the system

Solve the following system: $$dx/dt=-.2(y-2)$$ $$dy/dt=.8(x-2)$$ This is what I have so far, but I got stuck.. $$\begin{eqnarray} dx/dt&=&-.2y-.4\\ x'&=&-.2y-.4\\ ...
2
votes
1answer
29 views

Necessary and sufficient condition for separation of variables to give all solutions.

Lets say we have a partial differential with derivatives of $y$ with respect to $x$ and $t$ is there a necessary and sufficient condition that must be obeyed by such an equation for the superposition ...
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0answers
12 views

Greens function for boundary value problem

Determine the Green's function for the boundary-value problem:` $xy''+y'=-f(x)$ , $y(1)=0$ , $\lim_{x\to0} |y(x)|<\infty$ `
2
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1answer
49 views

Find $dy/dx$ of $(xy^2)+5 = x + 2y^2$

For the solution I got $$\frac{y^2-1}{ 4y-2xy} = dy/dx$$ I just want to know if this is correct. Also it says to evaluate $dy/dx$ at $(1,2)$. Would the solution to that be $3/4$?
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3answers
50 views

Determine the Taylor expansion for the solution of the differential equation

I'm given the following: $$\begin{cases}\frac{dx}{dt} = t^2x\\ x(0) = 1\end{cases}$$ I'm asked to determine the taylor expansion for the solution to the $t^{10}$ term. $$x(t) = a_0 + a_1 t + a_2 ...
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1answer
36 views

$x^{\frac{2}{3}}$ is not locally lipschitz at $x=0$

In order to show $x^{\frac{2}{3}}$ is not globally Lipschitz I can say that $\frac{u^{\frac{2}{3}}-0}{u-0} \rightarrow\infty$ as $u \rightarrow 0$ However, why does this tell me that ...
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1answer
26 views

Solving 2nd Order ODE w/Laplace Transforms + Heaviside

This is a similar problem to the one I posted earlier with some differences. Attempt at solution: Write g(t) as a heaviside function. Take Laplace transform of LHS and RHS. Solve for Y. Take ...
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0answers
19 views

Verify Solution inhomogeneous differential equation.

i'm doing a problem that should be handed in tomorrow. One of the problems is a differential equation and i'm a bit vague(right word?) on these types of problems.I'll show all the steps i've done. We ...
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1answer
27 views

First Order ODE – A skydiver weighing 180 lb falls vertically downward

...from an altitude of 5000ft and opens the parachute after 10s of free fall. Assume that the force of air resistance, which is directed opposite velocity, is of magnitude 0.75|v| when the parachute ...
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1answer
44 views

ODE arising in physics

I was solving a physics excercise that basically was about considering an object being gravitationally pulled from earth during a given time, but considering the variation of gravity along the way (it ...
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1answer
21 views

Harmonic oscillator differential equations

I need help finding the equations of the solution for which y(0)=-2 and v(0)=0 The differential equation was: 2d^2y/dt^2+6dy/dt+9y I rewrote the equation as a quadratic and got my general solution ...
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0answers
39 views

System of two ODE

I need to solve $$ \alpha F(x, y_0) = G(x) + y_0 + F_x(x, y_0)(H(x, F(x, y_0)) + \lambda_0 (F(x, y_{1}) - F(x, y_0)) \\ \alpha F(x, y_1) = G(x) + y_1 + F_x(x, y_1)(H(x, F(x, y_1)) + \lambda_1 (F(x, ...
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3answers
87 views

Nonlinear second-order ODE $yy'' - (y')^{2} = y^4$

I have the following ODE to solve. $$ yy'' - (y')^{2} = y^4 $$ I tried to substitute $y'$ by $v$, and then I get the following: $$ yv' - v^{2} = y^4. $$ I can't go further. I can't see what I'm ...
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1answer
31 views

Implications of a Gronwall-type inequality

Assume that $$f(t) \le K\int_a^t f(s)\, ds, \qquad\text{for all $\,t \in [a,b]$.} $$ for some constant $K$, where $f:[a,b] \rightarrow [0,\infty)$. Let $U(t) = K\int_a^t f(s) ds$. If $U(a) = 0$ and ...
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1answer
27 views

Picard theorem for functions which are locally lipschitz

I am a bit confused by the 4th bullet point. Is it just saying that $f$ is evaluated at points of $U$ for the time interval $t$?
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0answers
11 views

Looking for an ODE problem involving a tower and structural vibration

I am a teaching assistant of an ODE course (my assignment is just grading according to the grading guideline...) and the Prof. asked me to help design a undergrad level ode problem involving a tower ...
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2answers
75 views

Eigenvalues and eigenvectors of an integral operator

We have the following integral operator $$ Ku(t)=\int_0^1 G(t,s)\, u(s)\, ds,\,\, u\in L^2[0,1], $$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq ...
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2answers
23 views

Determine whether the solutions are stable or unstable.

Determine whether the solutions x(t)=0 and x(t)=1 of the single scalar equation $dx/dt=-x(1-x)$ are stable or unstable. So far in the book I have just done problems like this except with dx/dt=(some ...
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1answer
21 views

Finding a positive definite function to apply Lyapunov's Stability Theorem

Prove that the zero solution of the equation $$\ddot{x} + (1-x^2)\dot{x} + x=0$$ is stable by using Lyapunov's Stability Theorem. An indeed straight forward question but in general hard to ...
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0answers
21 views

Consider the equation: $x' = f(t,x)$. Prove that there is a two-way correspondence between the initial and the limits of the solutions.

Consider the equation: $$x' = f(t,x)$$ wherein, $$|f(t,x)| \leq \phi(t)x, \forall(t,x) \in \mathbb{R}\times \mathbb{R} $$ $$ \int^{\infty}\phi/(t)< \infty $$ If in addition, $f$ satisfying: $$ ...
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0answers
14 views

ODE and time: Short-hand notation?

I'm starring at this equation, trying to make sense of it. $$ a F(x) = S(x) + F_x(x)\cdot g(x, F(x))$$ I've seen this explained intuitively, but I'm having problems understanding it mathematically. ...
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2answers
53 views

How to calculate the errors of single and double precision

We consider the initial value problem $$\left\{\begin{matrix} y'=y &, 0 \leq t \leq 1 \\ y(0)=1 & \end{matrix}\right.$$ We apply the Euler method with $h=\frac{1}{N}$ and huge number of ...
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1answer
12 views

How do I see that a homeomorphism $\sigma$ is an open function?

How do I see that a homeomorphism is an open function ? Given a homomorphism $\sigma: X \rightarrow Y$ between topological spaces, how do i then see that $\sigma(V)$ is open in $Y$ for $V$ open in ...
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0answers
36 views

Complex Exponential in Differential Equations.

I am a physics student, but have taken courses in analysis and algebra. My knowledge of differential equations is purely methodical, and I was hoping for a more math oriented insight with regards to ...
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1answer
20 views

Using Exponential Shift to find general solution of an ODE

We have these 2 theorems/definitions. *For each natural number n, $(D-m)^n e^{mx} y = e^{mx}D^ny$ *If f(D) is a polynomial in D with constant coefficients then ...
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1answer
21 views

Find all equilibrium values of the given system of differential equations:

Find all equilibrium values of the given system of differential equations: a. $$\frac{dx}{dt}=x-x^2-2xy$$ $$\frac{dy}{dt}=2y-2y^2-3xy$$ b. $$\frac{dx}{dt}=-\beta xy+ \upsilon$$ ...