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

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How to form the equation of a line from a gradient?

I am given that the gradient of a curve is $dy / dx = 10x^4 - 6x^2 + 5 $ And I need to find the equation of the curve. I started by integrating this (as it is the reverse of differentiation) and got ...
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41 views

Unique solution?

If I have the function $f \colon \mathbb{R} \to \mathbb{R}$ with $ f'(t) = k \cdot f(t) $ how can I argue that this solution has to be of the form $f(t) = Ce^{kt} $ and can't look any different? ...
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91 views

Need help on books on diff. equations/geometry and theoretical computer science

I am looking for recommendation of 3 different books on the following topics: 1.Differential Equations -Ordinary diff. equations -Vector field, transport equations -Equation of wave and heat -Use ...
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57 views

The Differential Equation $\frac{dy}{dx}=60(y^2)^{1/5}$

The Differential Equation $$\frac{dy}{dx}=60(y^2)^{1/5}$$ $x>0$, $y(0)=0$, has a unique solution, two solutions, no solution, infinite number of solutions. After ...
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128 views

Nonlinear first order system of ODEs

While solving some physical problem, I have obtained the following system of differential equations with boundary conditions: $$\left\{\begin{matrix} \frac{d\phi_1}{dz}=\frac{m^2}{\lambda}- ...
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24 views

Interval of Initial Value Problems

The problem states: $y= 1 / (x^2 + C)$ is a solution to $y' + 2xy^2 = 0$ if $y(2) = 1/3$, find $C$: Simply enough, $C = -1$. But now it says, Give the largest interval I over which the solution is ...
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63 views

integro-differential equation with application in quantum mechanics

I am trying to solve for the time dynamics for a simple quantum system (two-site system with sinusoidal coupling and a decay parameter on one site) and the math is looking not so simple. Here is the ...
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34 views

How does this wlog-argument work?

Let $G\subset\mathbb{R}^n$ be open, $f\colon\mathbb{R}\times G\to\mathbb{R}^n$ continious and locally Lipschitz-continious in $x$, consider the IVP $$ ...
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87 views

System of Nonhomogeneous DEs - Help Solving???

I'm studying for finals at the moment and could use some help with solving the particular solution for this system of nonhomogeneous differential equations: $x' = \begin{bmatrix}1 & 0\\ 2 & ...
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136 views

Reference request: Gronwall's inequality with negative sign(s)

The following claim is a consequence of Gronwall's theorem Let $x \colon [0,\infty) \to \mathbb R$ with $x(0) = 0$ be a continuously differentiable function, whose derivative satisfies $$ \dot ...
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4answers
246 views

Mathematical modelling that involves projectile motion

I was asked to solve a mathematical differential equation to find the time taken by an object to reach the highest point and the time taken by the object to fall from its highest point to ground. I ...
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46 views

Finding general solution of first order DE's using integrating factor

I am asked to find the general solution of $$R\frac{dq(t)}{dt}+\frac{q(t)}{C}-V_0=0$$ I re-arrange so it is in the correct format. $$\frac{dq(t)}{dt}+\frac{1}{CR}\cdot{q(t)}=\frac{V_0}{R}$$ ...
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How to solve $x=tx'^2+x'^3$

How about the differential equation $x=tx'^2+x'^3$. Can somebody solve it? my solution:suppose $\dot x$=p,then $x=tp^2+p^3$. Then, p=$p^2$+2tp$\dot p$+3$p^2\dot p$, then we can use these function to ...
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1answer
154 views

Jacobian linearization, does it need to be around a hyperbolic fixed-point?

Everything that I read about Jacobian linearization of systems of nonlinear equations is about approximations near hyperbolic fixed-points (cf. the Hartman-Grobman theorem). It seems to me that even ...
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41 views

Converting a Differential Equation into a system

What is the procedure for converting a single differential equation into a system to then be solved by matrix methods. I've looked it up on a few websites but I still don't understand what is being ...
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1answer
85 views

Laplace transform of piecewise continuous function

$$f(t) =\begin{cases}t^2 & 0 \le t < 3,\\ 9& t \ge 3\end{cases}$$ Show that $f$ is of exponential order. Express $f$ in terms of the unit step function. Find Laplace transform of ...
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Gompertz growth model problem

The growth of tumor cells is characterized with Gompertz model. $N'=-aNln(bN),$ where N(t) is proportional to the number of cells in the tumor, while a and b denote positive parameters. ...
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172 views

Drawing the trajectories for a non-linear system

Find the critical points of the system and draw the trajectories, indicate if they are stable, asy. stable or unstable. $dx/dt=x(1.5-x-0.5y)\\dy/dt=y(2-y-0.75x)$ My critical points are $(0,0) ...
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28 views

Non-homogenous System where did I go wrong?

Solve the system $\vec{x^{'}}=\begin{pmatrix}2 & -5\\1 & -2 \end{pmatrix}\vec{x}+ \begin{pmatrix} -\cos t\\ \sin t \end{pmatrix}$ The Eigenvalues are $(2-\lambda)(-2-\lambda)+5=0 \implies ...
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93 views

How to plot not only the result but also the derivatives of an ode using the ode45 function in Matlab?

I have already successfully run a code for the simulation of the deflection of beams under different loadings. I used the Matlab program and the ode45 solver for Initial value Problems and bvp4c ...
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27 views

Non-Homogenous System answer not matching

Find the Genereal Solution of $\vec{x}^{'}=\begin{pmatrix}2&-1\\3&-2 \end{pmatrix}\vec{x}+\begin{pmatrix}1\\-1 \end{pmatrix}e^t$ I found the eigenvalues to be $\lambda=\pm 1$ Therefore the ...
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46 views

bifurcation with more than parameter

Problem: Consider the scalar differential equation depending on the parameters $\alpha_1, \alpha_2$ ∈ $\Re$ $x˙ = \alpha_1 + \alpha_2 x − x^2$. Find a change of coordinates $y = \phi(x)$ such that ...
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418 views

Solve second order differential equation with Heaviside function using Laplace transform

The equation is: $$y'' + 3y = u_4(t)\cos(5(t-4)), \quad y(0) = 0, \quad y'(0) = -2$$ Here $u_4$ is the Heaviside function with activation switch at $t=4$. I can get all the way to the partial ...
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63 views

Question about Poisson formula

We have the Laplace equation in polar coordinates: $$u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta \theta}=0, 0 \leq r <a, 0 \leq \theta \leq 2 \pi$$ With the separation of variables, the solution ...
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95 views

Laplace’s equation in the Polar Coordinate System

Laplace’s equation in the Polar Coordinate System: ...
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61 views

Using Laplace Transform to solve a equation with piecewise function

Using Laplace Transform to solve$$y''+4y=f$$ Where $y(0)=0, y'(0)=-1,$ and:$$f(t)=\begin{cases}\cos(2t)&\text{if $0\le t \lt \pi$}\\0 &\text{otherwise}\\\end{cases} $$ Do I need to solve the ...
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30 views

Laplace transform using the definition

Find the Laplace of the given function using the definition $$f(t)=tsin(t)$$ I know what the answer is according to a sheet that I have of common transforms but I am not 100% on how to get there ...
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146 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
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41 views

Problem with checking whether $x(t)$ can be a solution of any system of first order homogeneous ODE

I need to find out whether $$x(t) = (3e^t + e^{-t}, e^{2t})$$ can be a solution of the system $$\dot{x} = A x\quad \quad (1)$$, where $A$ is a $2x2$ matrix. I'm not sure of my solution, which is the ...
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239 views
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Uniqueness of the solution to some differential equation.

I'm currently working on the subject mentioned in the title in a very general way. I think I get stuck for a stupid reason but here is my problem : I'd like to show that any solution to the equation ...
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48 views

Find equation of the curve

The product of the slope of the tangent line to a curve and sum of the coordinates of the point of contact equal to the ordinate for any point of the curve. This curve pass through the point $M_0 ...
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53 views

Bifurcation flow field?

assuming I have $$x'=5+mx+2x^2$$ how would I find the flow field of this bifurcation with the changing variable m? EDIT: Take for example ...
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111 views

Solutions for an ODE

I am looking for a solution of the ODE $x'(t)=x(t)+\frac{1}{1+e^{t}}$ which has finite limit when $t\rightarrow \infty$, I already find that the solutions are $e^t \ln(1+e^t)-te^t-1$ however these ...
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Need some help with a second-order non-linear differential equation

I came across this differential equation in a problem I'm working on: $$m \ddot{r} - \frac{a}{r^3} + b =0 \, ,$$ where $m$, $a$ and $b$ are positive constants and $r=r(t)$. Furthermore, since this ...
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235 views

Legendre trigonometric form

Consider the Legendre equation for a function $y(x)$ defined in the interval $-1 < x < 1$ By a change of variable $x = cos \theta$ derive the trigonometric form of Legendre equation for a ...
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54 views

Erratic outputs from a second order system when input is too small

I'm modeling a simple Mass-Spring-Damper system to represent the torsional behavior of a micromirror. With references to some papers (this one mainly), I've constructed the model of a torsional mass ...
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79 views

Proof that Lipschitz condition guarantees well posedness of initial value problems

In the proof of the theorem which states that the Lipschitz condition guarantees well posed-ness of an initial value problem $y'=f(x,y)$, $y(x_0)=y_0$, I came across this Let the perturbed problem be ...
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72 views

Traffic flow vs Density

This is a pretty simple question but I can't seem to understand it conceptually. The question is: If the traffic flow is increasing as $x$ increases ($\frac{\partial q}{\partial x}>0$), explain ...
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79 views

equilibrium solutions question? with initial value and limit question?

$y'=y(y-2)(y-3)^2$ if $y(0) = 1$ what is the limits of $y(x)$ as $x \to +/-\infty?$ I have found the solutions of the differential equation and sketched the direction field if thats to any help on ...
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28 views

find the max. value of the area A(t)..

For $0<t≤1$ ,let $A(t)$ denote the area of the triangle bounded by the x-axis, the y-axis and the tangent line to the curve $y= \ln x$ at $(t,\ln t)$. Find the maximum value of area $A(t)$.
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Linear Phase Plot

In my text book I have the following question (it's a dutch text book, so I hope I translate the mathematical terms correctly) Draw a Phase Plot of the following system: $\frac{dy}{dx}=A_i y$ With: ...
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36 views

Approximation of the solution of an IVP

Consider the initial value problem $$\frac{dy}{dx} = x^2 + y^2, \\ y(0) = 0$$ on D = {|x| <= 1, |y| <= 1} Find the third approximation to the solution If someone could maybe walk me through ...
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32 views

How can I solve this differential equation through Laplace transforms?

Solve $y''-2y'+2y=0$ where $y(0)=0,~~y'(0)=1$. So I started solving this by doing the following $$s^2Y(s)-sy(0)-y'(0)-2sY(s)-y(0)+2Y(s) = 0 \\ (s^2-2s+2)Y(s)-1=0 \implies Y(s) = \frac{1}{s^2-2s+2} = ...
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114 views

Solving $x^2y''-xy'+y=x,\;x>0$ with non-constant coefficients using characteristic equation?

$$x^2y''-xy'+y=x,\;x>0$$ Whenever you deal with non-constant coefficients you usually use Laplace transform to solve a given differential equation, at least that's how how I learned it. How to ...
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35 views

Finding the General Solution to the system of equation

Find the General solution of $\textbf{x}^{'}=\begin{pmatrix} 2&2+i\\-1&-1-i\\ \end{pmatrix}\textbf{x}$ I started out by finding the eigenvalues. ...
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52 views

Solving ODE involving piece-wise function

This question may sound very trivial but I really can't figure out how to solve this. It has been a while since I solved ODEs and I need to solve an equation similar to one given below for some ...
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67 views

solve partial differential equation

$$y^2u_x + xu_y = \sin(u^2) \\ u(x,0)=x$$ I get the projected characteristic curve on xy plane easily. However, cannot get the other one. actually the problem is getting the value of $U_{xx}, ...
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151 views

Having trouble integrating for use of energy method to prove uniqueness

We are given $u_{tt} - c^2u_{xx} + ru_t$. To prove only one solution exists, I am taking w = $u_1 - u_2$, assuming they are both solutions to the given wave equation. So: $u_{tt} - c^2u_{xx} + ...
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Why was this change of variables made in this system of differential equations?

I'm trying to understand an example from my notes. I'm given a system of linear differential equations as follows $$x'=2x-y$$ $$y'=2x-2$$ The notes solve these by making the change of variables ...