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

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Help solving an ODE

This is an example in my book. It is for the following system: \begin{align*} x'&=y+x(1-x^2-y^2)\\ y'&=-x+y(1-x^2-y^2) \end{align*} So using polar coordinates we get the following system ...
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What about uniqueness of general solution?

I found some info about uniqueness for inital value problem. But what about uniqueness of general solution? Is it right that ODE $y'=y$ has two general solutions? 1) $y=Ce^x$ 2) $y=e^{(x+C)}$ Or ...
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How to find I(t)?

(1) $I'=\frac{dI}{dt}=\beta SI=\beta(K-I)I$ , since $S=K-I$ Then, $\frac{dI}{I(K-I)}=\beta dt$. I know how to find I(t) with this differential equation using partial fractions. How can I find I(t) ...
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1answer
52 views

Best Book For Differential Equations?

I know this is a subjective question, but I need some opinions on a very good book for learning differential equations. Ideally it should have a variety of problems with worked solutions and be ...
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9 views

show uniform convergence of picard iteration

Let $\frac{d}{dt}u(t)=tu(t)+t^3$ and $u(0)=0$ with $u:\mathbb R\rightarrow\mathbb R$. Given the following iteration of Picard $u_{k+1}(t)=\int_0^t f(s,u_k(s))ds$ with $u_0=0$ how can you find ...
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Modification of Gronwall's Lemma

Exercise 2.3 in this book: ...
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Proof of Gronwall's Inequality

I have a question about the proof of Gronwall's inequality as given in Chicone: Ordinary Differential Equations with Applications. Gronwall: Suppose that $a<b$ and let $\alpha, \phi,$ and $\psi$ ...
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How do impulsive differential equations work? Can you provide an example?

I have heard of impulsive differential equations being used in some epidemiological models of infectious disease. I haven't heard of them before in my math education, and I was wondering how they ...
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Existence of a solution of a nonlinear ODE

I have to show, that the nonlinear ODE $$u'(t)-2u''(t) u(t)=-1,\quad u(0)=1,\,u'(0)=0$$ has a unique solution $v(t)\in C^2(0,T)$ on any Interval $[0,T]$, $T>0$ and that $$\max_{0\leq t\leq ...
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1answer
14 views

Checking the solution to a diffential equation.

Is there a quick way to check that the solution to a diffential equation is correct, I know you can diffentiate it and see if it works but this can take a long time (I want to check my answers in an ...
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1answer
33 views

Finding the Green's function for $y'' + y' = f(x)$

I have this ODE: $$y'' + y' = f(x)$$ with $y(0)=0$ and $y'(1) = 0$. I'm trying to find the Green's function. I multiply through by $G$, integrate over the domain and then use integration by parts to ...
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Are there other well known oscillator systems besides Van der Pol oscillator? [on hold]

Is there any collections of oscillator systems similar to "matrix market"?
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20 views

Which $n$th order differential equations have $n$ linearly independent solutions?

In these notes (p. 28), it is stated that differential equation $28$ is a second order ordinary differential equation therefore there are two linearly independent solutions. Which is the largest set ...
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2answers
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the global stable and unstable manifolds

Show that $x^* = (1, 2)$ is a fixed point of the system $x_1' = 2 + 3x_1 − 2x_2 − x_1^2 + 2x_1x_2 − x_2^2$ $x_2' = 3 + 4x_1 − 3x_2 − x_1^2 + 2x_1x_2 − x_2^2$ Determine $W^s(x)$ and $W^u(x)$, the ...
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2answers
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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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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 (ii) and the rest. ...
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Laplace Transform of an integral

Find the Laplace transform of $$f(t)=t\int_0^{t} \tau e^{-\tau}$$ $L(f)(s)$= ?? My thought is that I can change the $\tau$ to $t$ by Transforming the integral to get $$t/s*L[t*e^{-t}]$$ But ...
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1answer
21 views

Laplace transform of integral equation

Use Laplace transforms to solve the integral equation $$y(t)-\frac{1}{2}\int_0^ty(t-v)~dv=1$$ First find the Laplace transform $Y(s)$ of $y(t)$
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Convolution of two equations

Find the convolution of $f(t) = t$ and $g(t) = e^{t}$ $$(f*g)(t)= ?$$ If I am correct, I am able to find the Laplace Transform of each individually, then multiply them together. Let $L(x)$ equal ...
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interval of solution of a Linear Ordinary Differential Equation with initial conditions

The equation is $$y' + \frac{2ty}{t^2-4} = \frac{2t}{t^2-4}$$ with $y(0) = 1$ as initial condition. What is the solution and its interval? Using some methods of solution I can come up with $y(t^2 - ...
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1answer
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Wronskian Bessel Equations

I need to compute the wronskian of $J_n$ and $Y_n$ (the Bessel functions of the first and second kinds). I've been able to find in many sources that it is $$W(J_n,Y_n)=\frac{\pi}{2x}$$, but I haven't ...
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Using polar form to show that a simple critical point is a spiral point

This is the question in my "homework." I say "homework" because it is not picked up or graded but we are supposed to do it for practice, anyhow here's the question: Given the system ...
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2answers
25 views

Laplace transform of multiplication of three terms

Okay, so I have $${f}(t)= t\mathrm{e}^{-2t}\sin 2t.$$ In order to do a Laplace transform, I'm pretty positive I cannot just split it up cause that would basically break the rules of math. I ...
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Show that the D'Alembert operator is a formally self-adjoint operator.

A problem asking me to prove that the D'Alembert operator, defined as: $$\hat\Box^2=\frac{\partial^2}{\partial t^2}-\bigtriangledown^2$$ is a formally self-adjoint operator. To demonstrate the ...
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1answer
28 views

Meaning of $ dx \times dy = k $

Does $ dx \times dy = k $ have a mathematical meaning? What about when considering $y = y(x)$?
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Question about an eigenvalue problem

I have a question... How can I show that the eigenvalue problem $$y''+λy=0$$ $$y(0)=0,$$ $$ y'(0)=\frac{y'(1)}{2}$$ is NOT a Sturm-Liouville problem?
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1answer
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Which means adjoint problem of a differential equation?

I wanted to know if anyone can help me with the following problem: Get the adjoint problem (differential equation and boundary conditions) for the problem given by: $$\frac{d^2 u}{dx^2}=f(x)$$ ...
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1answer
25 views

Good Source of Differential Equations Problems with Worked Solutions?

I am looking for a good source of problems for differential equations (first order, second order, laplace, convolution, systems). I find it helpful if the question has a worked solution or at the ...
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7 views

Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
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1answer
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Stability of solution to first-order nonlinear differential equation

The problem is to consider $u'(t)+u(t)=\cos(u(t))$ posed as an initial value problem for $t>0$ with initial condition $u(0)=u_0$. The first part asks to show that there is exactly one solution $u$ ...
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Help Finding Critical Points of a Cubic (with 2 parameters)

I am trying to find bifurcation points in 1 dimension, but am having trouble finding critical points of $x'=\mu x -2x^2-x^3+ \delta$ ( where $x$ is my variable, $\mu$ is a parameter, and $\delta$ ...
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Bessel Functions Proof

How would I even begin to start proving the following? After looking at Frobenius' method and the Rieman P-equation, I started delving into the derivation of Bessel's/Legendre's functions, and I ...
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1answer
34 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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How to solve Cauchy problem?

I'm new to this problem. Here is the question. $$(y+xz)z_x+(x+yz)z_y=z^2-1$$ Find the integral surface which the curves it passes are $y=1$ and $z=x^2$ By Lagrange system i found $u$ and $v$. We ...
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1answer
53 views

Solution to ODE Abel Equation

I aim to find the exact form solution to the this ODE: $$\frac{dS}{dw}S = \frac{a}{w}S^2 + \frac{b}{w}S - c$$ where S is a continuous differentiable function of w, real positive and a, b, c are ...
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predictor-corrector method and stability

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
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Finite difference scheme and its stability

The Finite difference scheme: \begin{equation} y_{n+3}-y_{n+1}= \frac {h}{3}(f_{n}-2f_{n+1}+7f_{n+2}) \end{equation} Deduce that the scheme is convergent and find its interval of absolute stability(if ...
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Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
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Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
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Forced harmonic oscillator [on hold]

I am completely stuck, any help or advice would be appreciated, big thank you. The motion of a forced harmonic oscillator can be described by the differential equation $$y''(t) + γ\,y'(t) + y = ...
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1answer
28 views

Solving a differential equation?

I'm trying to analyze the transient state of a RC circuit. My book gives me the following differential equation: $$\frac{d(v(t))}{dt} + av(t) = c$$ for some constants $a$ and $c$. The book thens ...
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defining ODEs recursively in maple

I want to numerically look at a system of ODEs with a large number of variables; defined by $da_j(t)/dt= 2^j a_{j-1}^2 - 2^{j+1} a_j a_{j+1}$, for $j=0\ldots50$ with $a_{-1}= a_{51}=0$. In maple, I ...
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3-Species Population Model

I am trying to solve a 3-species predator-prey system in matlab. Here is the equation: $$\frac{d}{dt} \begin{bmatrix} N_1 \\ N_2 \\ N_3 \\ \end{bmatrix} = \begin{bmatrix} N_1 & 0 & 0 \\ 0 ...
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Demonstrate identity $\int_{0}^{x} \int_{0}^{\xi}f(s)dsd\xi=\int_{0}^{x}(x-\xi)f(\xi)d\xi$

I have trouble doing the following problem, I have not been able to make even the first part, I was hoping someone could help me. The problem is: Show that: $$\int_{0}^{x} ...
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1st order differential equation

I am given the following: $$ \begin{cases} x \ln x \frac{dy}{dx}+y + x = 0, &\mbox{if}\quad x>1, \\ y = 0, &\mbox{if} \quad x=e \end{cases} $$ I tried to separate it and got this: $$ -y \ ...
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$w''+w~ sin z=0$ solutions

I have $w''+w~\sin z=0$ and I want to sow that this equation has at least one solution of the form $e^{c z}v(z)$ where $v(z)$ is periodic My idea is to start by expressing $w_1(z+2\pi)$ and ...
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2answers
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How to solve $y''+9y=-18\sin{3x}-18e^{3x}$?

Here is my solution so far: $$y''+9y=-18\sin{3x}-18e^{3x}$$ 1.Find complementary soultion.$$y''+9y=0$$ assuming that solution will be in form $e^{kx}$, substitute $y=e^{kx}$, $$k^2e^{kx}+9e^{kx}=0$$ ...
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second order linear ode in the complex domain

Consider $w''(z)+p(z)w'(z)+q(z)=0$ where $p(z), q(z)$ are analytic for $R\le|z|<\infty$ for some fixed $R$. Now I want to prove using analytic continuation of the solutions that the ode has one ...
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Solve by separating variables

$$\frac{dy}{dt}=e^y +1$$ I've tried: $$dy/dt - e^y = 1 $$ $$\Leftrightarrow y' - e^y dt = 1 dt$$ But I'm not sure what to do next or if I'm even doing this right!