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

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3
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1answer
403 views

Explain the error term in Euler method

Task: I had to find out some estimates for M and L to make sure the proportional accucrazy is not above $10^{-4}$ in the Euler method with the problem below. I am trying to understand the page 672 on ...
3
votes
2answers
249 views

What is the formal definition of $d$, or $\partial$, in differation and integration

This might sound a bit like a silly question, but i'm a second year math student, and so far i've encountered $d$ or $\partial$ in many cases ofcourse (mostly in calculus :)). Those letters or symbols ...
2
votes
1answer
72 views

Stability analysis for a system of two differential equations

I have this system of differential equations: \begin{equation} \frac{dx}{dt}=\alpha x-\beta xy\\ \frac{dy}{dt}=\beta xy-\gamma y \end{equation} I need to find the critical points and then do a ...
2
votes
1answer
74 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
2
votes
1answer
54 views

Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ z z' $?

This is a follow-up to Under what conditions can a function $ y: \mathbb{R} \to \mathbb{R} $ be expressed as $ \dfrac{z'}{z} $?. It turns out that in that case, \begin{align} \text{$ y = ...
2
votes
0answers
56 views

modified ODE has same trajectories as original system and associated flow is defined for all $t \in \mathrm{R}$ [closed]

I really don't know where to start with this problem. Consider the differential equation $\dot{x} = f(x)$ with $f \in C^1(\mathrm{R}^n,\mathrm{R}^n)$. Consider the following modified differential ...
2
votes
2answers
160 views

Finding the Asymptotic Curves of a Given Surface

I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$. I guess that what was meant by that statement is that surface ...
2
votes
2answers
122 views

Solve the following differential equation: $ty' + 2y = \sin(t)$

An exercise from the book: Solve the following differential equation: $ty' + 2y = \sin(t)$ This is the first time I approch a differential equation, and the book doesn't provide an example how ...
2
votes
2answers
238 views

Third order ODE initial value problem,solution obeys $y(x) \rightarrow 0 $ as $x \rightarrow \infty$ ???

$y''' + y'' -y' -y=0$ $y(0)=7,y'(0)=-3,y''(0)=\alpha$ Find all values of $\alpha$ for which the solution obeys $y(x) \rightarrow 0 $ as $x \rightarrow \infty$ Here is my work I used the cubic ...
2
votes
1answer
2k views

Polar coordinates differential equation

I have the following ODE: $$\dot x=-y(x^2+y^2), \dot y=x(x^2+y^2)$$ I want to sketch the phase portrait (manually) and I want to find the flow $\phi_t$, the orbit $O(x_0)$ and the limit set ...
2
votes
2answers
103 views

On the existence of a particular solution for an ODE

The problem asks to find a bounded $u(\cdot) \in \mathcal{C}^2(\mathbb{R})$ such that $$u''+u'-2u=f$$ where $f$ is a bounded continuous function on the real line. [Observations, Editted] We can ...
2
votes
1answer
353 views

Stability of nonlinear system with borderline linearization

I have the following nonlinear system: \begin{align} ...
2
votes
3answers
115 views

Getting equation from differential equations

I have: $\dfrac {dx} {dt}$=$-x+y$ $\dfrac {dy}{dt}$=$-x-y$ and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$ I know to do this I need to decouple the equations so that I ...
2
votes
1answer
108 views

Solving $f_n=\exp(f_{n-1})$ : Where is my mistake?

I was trying to solve the recurrence $f_n=\exp(f_{n-1})$. My logic was this : $f_n -f_{n-1}=\exp(f_{n-1})-f_{n-1}$. The associated differential equation would then be $\dfrac{dg}{dn}=e^g-g$. if ...
2
votes
1answer
766 views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
2
votes
2answers
696 views

Solving a 2nd order differential equation by the Frobenius method

Can you, please, help me to solve this equation: $$(x+1)^2y''+(x+1)y'-y=0$$ Here, for me the problem is, I am finding relationship among 3 members: $a_n, a_{n+1}, a_{n+2}$, not between 2 members: ...
2
votes
1answer
525 views

Show that Bessel function $J_n(x)$ satisfies Bessel's differential equation.

here is the question: For each positive integer $n$, the Bessel function $J_n(x)$ may be defined by $$J_n(x) = \frac{x^n}{1\cdot 3\cdot 5\cdots(2n-1)\pi}\int^1_{-1}(1-t^2)^{n-1/2}\cos(xt) \, dt$$ ...
2
votes
2answers
530 views

Clarification of Frobenius method roots

The frobenius method states that for repeated roots or roots that differ by an integer, an alternative method must be used to find the second solution once one is found. When they say "roots that ...
2
votes
2answers
936 views

second derivative of the inverse function

I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = ...
2
votes
2answers
159 views

Inhomogeneous equation

Let $A$ be an $n\times n$ matrix and $\beta$ a constant. Consider the special inhomogeneous equation $$\dot x = Ax + p(t)e^{\beta t},$$ where $p(t)$ is a vector all whose entries are polynomials. Set ...
2
votes
1answer
100 views

The system $x'=Ax$ is an attractor if and only if there is a positive quadratic form q such that $Dq(x)\cdot A(x)<0$ for all x

I need to show this result: Given the system of ODEs $x'=Ax$, the origin, $0$, is an attractor (equivalently, all the eigenvalues of the real matrix $A$ are negative) if and only if there exists a ...
2
votes
1answer
225 views

To find the closed form of $ f^{-1}(x)$ if $3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$

$$3f(x)=e^{x}+e^{\alpha x}+e^{\alpha^2 x}$$ where $\alpha=e^{\frac{2\pi i}{3} }$ I would like to find a closed form of $ f^{-1}(x)$ $$f(x)=\sum \limits_{k=0}^\infty \frac{x^{3k}}{(3k)!}$$ We can ...
2
votes
2answers
1k views

Use of Legendre's equation.

For some weeks have been studying Legendre polynomial as a solution to this equation. $$ (1-x^2)\frac{d^2}{dx^2}f(x)-2x\frac{d}{dx}f(x)+n(n+1)f(x)=0.$$ I've found them very interesting to learn from ...
2
votes
2answers
553 views

Help on differential equation $y''-2\sin y'+3y=\cos x$

$y''-2\sin y'+3y=\cos x$ I'm trying to solve it by power series, but I just can't find the way to get $\sin y'$. Is there any special way to find it?
2
votes
1answer
419 views

Existence of global solution of Riccati equation

Consider a Riccati differential equation $$ \dot P + A(t)^{T}P + PA(t) -PB(t)R(t)B(t)^{T}P + Q(t) = 0,\;\;\; P(t_0) = P_0 = P_0^{T} \geqslant 0 $$ where $Q(t) = Q(t)^{T} \geqslant 0$, $R(t) = ...
2
votes
4answers
1k views

$y'''-y=x^{2}$ has solution — `“multiplicity”`?

The page 667 of the book (sorry not in English) claims $y'''-y=x^{2}$ to have the solution $$y(x)=C_{1}e^{x}+e^{-x/2}\left(C_{2} \cos \left( \frac{\sqrt{3}x}{2} \right)+C_{3} ...
2
votes
3answers
115 views

Some double angle identity to solve $(2x^{2}+y^{2})\frac{dy}{dx}=2xy$?

For some reason, I cannot see a clever way to solve this (I know the way of doing it like in Wolframalapha) but I am pretty sure there is a double angle identity to crack this puzzle. Could someone ...
2
votes
3answers
149 views

The equation $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$

Can one give a hint how to solve the following equation? $(x-2xy-y^2)\frac{dy}{dx}+y^2=0$ Thanks in advance.
2
votes
2answers
488 views

Poincare-Bendixson Theorem

Can someone sketch some ideas of how to use the Poincare-Bendixson Theorem to prove that there must be a fixed point contained inside a periodic orbit?
1
vote
2answers
48 views

Linear differential equation and its Wronskian

Let $a(t),b(t)$ be continuous functions and $x_1(t),x_2(t)$ two solutions of the differential equation $$x''(t)+a(t)x'(t)+b(t)x(t)=0$$ We define $w(t)=x_1(t)x_2'(t)-x_2(t)x_1'(t)$. Show that (i) ...
1
vote
1answer
45 views

Differential equation of falling object

Suppose a mass is dropped from a height of 300 m, whose speed obeys the differential equation $$\frac {dv}{dt} = 9.8 - \frac v5 . \tag{1}$$ We want to find the time of and speed at impact with the ...
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2answers
84 views

Show that the parameterized curve is a periodic solution to the system of nonlinear equations

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the ...
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3answers
54 views

Solving an ordinary differential equation with initial conditions

Can someone please help me with this ODE problem? Here is the question: Consider the ODE $ {d^2 U\over dx^2} - [{s^2\over c^2}]U=e^{{-sx\over v}}. U(0) = 0, U(x)$ is bounded as $x$ goes to ...
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4answers
100 views

Separable First Order Ordinary Differential Equation with Natural Logarithms

Please check my work: $$xy' = 5y$$ $$\int\frac{dy}{y} = 5\int\frac{dx}{x}$$ $$\ln y = 5\ln x + c$$ $$y = 5x + c$$ Is this correct?
1
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1answer
54 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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1answer
49 views

Second order differential equation with non constant coefficients

I want to solve the following differential equation: $$ 2f'(x)(2x+1)+\frac{\kappa}{2}f"(x)x(x+1)=f(x)(\frac{-2b}{x+1}+\frac{2c}{x}+2a) $$ where $\kappa,a,b,c$ are arbitrary positive constants. Is ...
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1answer
40 views

Let $y'' + p(x)y' + q(x)y = 0$ , where $p(x)$ and $q(x)$ are continuous. Prove that the zeroes of $y$ are isolated.

Let $p$ and $q$ be continuous, and let $y$ be any solution of $y′′(x) + p(x)y′(x) + q(x)y(x) = 0$ that is not identically zero. Then zeroes of $y$ are isolated, in the precise sense that for any ...
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1answer
64 views

Interpreting another proposition full of symbols

Could someone help me interpret the following proposition full of symbols? I've been struggling to comprehend it. Thanks in advance. Proposition: Suppose that $f:\mathbb{R^n} \rightarrow ...
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4answers
210 views

Inhomogeneous 2nd-order linear differential equation

I need to solve this: $y'' + ay' + by = x(t)$ where nothing about the form of $x$ is known, except that it is bounded and non-negative. In addition it is known that $y(0) = 0$ and $y'(0) = 0$ (and ...
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1answer
109 views

Question about an O.D.E

I have this theorem: Suppose that $U$ is a neighborhood of $\theta$ in a Hilbert space $H$ and that $f\in C^2(U,\mathbb{R}^1)$. Assume that $\theta$ is the only critical point of $f$ and that ...
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0answers
63 views

Inverse Laplace Transform using Jordan's Lemma?

Following is the question that i am trying to solve: "Consider a second order linear ODE $x\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+(3-2x)y=0$ A) Find the solution employing Laplace integrals by ...
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1answer
85 views

Counter-example to Cauchy-Peano-Arzela theorem

I was looking for a counter-example to Cauchy-Peano-Arzela theorem. I found this paper (in french) from Dieudonné. [acta.fyx.hu] Take $E = c_0$ to be the space of real sequences converging to $0$, ...
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1answer
510 views

Prove Friedrichs' inequality

I'm trying to show that the theorem (Friedrichs' inequality) in my book: Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the ...
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0answers
23 views

Getting Eigenvalues Into a Differential Operator?

Following Butkov, a second order ode $$A(x)y'' + B(x)y' + C(x)y = D(x)$$ can always be brought into Sturm-Liouville form $$\tfrac{d}{dx}[p(x)y'] - s(x)y = f(x)$$ after multiplying across by ...
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1answer
220 views

Show that a differential equation satisfies Lipschitz condition

Prove that if $$\frac{dx}{dt}=(3t^2+1)\cos^2(x)+(t^2-2t)\sin (2x)=f(t,x),$$ then $f(t,x)$ satisfied Lipschitz condition on the strip $S_{\alpha}=\{(t,x):|t|\le\alpha , |x|\le \infty , \alpha >0\}$. ...
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2answers
173 views

Understanding differentials

What is a good reference to learn about differentials and related topics. Some of my questions are: Why is it possible to split $dy/dx$ into individual terms $dx$ and $dy$? In a separated ...
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0answers
71 views

When is it justified to approximate a difference equation with its corresponding differential equation?

Consider the difference equation $f_{x+1}-f_x=a(f_x)$ and the differential equation $g'_x=a(g_x)$. When and Why is it justified to say "$f_x - g_x = o(1) $ hence we can solve the difference equation ...
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2answers
233 views

resources to study PDE from

I am an undergrad engineering student. I recently completed my second year, with that said, I have taken several calculus courses. Most recently I completed differential equations and multivariable ...
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1answer
442 views

Nonlinear phase portrait and linearization

Consider the nonlinear system $x^{'} = y$, $y^{'}= -8 \sin x - 2y$ where $-2\pi$ < or = x < or = $2\pi$ Find the equilibrium points of the system. $(-2\pi,0)$$(-\pi,0)$$(0,0)$ ...
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1answer
502 views

System of differential equations with triple eigenvalue

Could anybody, please, explain to me, how to solve system of 3 differential equations, when it has triple eigenvalue? I mean... we solved these equations by creating a matrix $A$ of the system and ...