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

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2
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2answers
217 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
1k 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
101 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
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1answer
334 views

Stability of nonlinear system with borderline linearization

I have the following nonlinear system: \begin{align} ...
2
votes
3answers
114 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
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1answer
739 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
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2answers
655 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
508 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
498 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
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2answers
884 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
99 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
224 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
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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
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2answers
549 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
408 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
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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
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3answers
113 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
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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
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2answers
483 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
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2answers
43 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
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1answer
43 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 ...
1
vote
3answers
53 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 ...
1
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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
52 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 ...
1
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1answer
45 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 ...
1
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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 ...
1
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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 ...
1
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4answers
209 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 ...
1
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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
62 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 ...
1
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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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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
213 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
161 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
213 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
424 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
487 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 ...
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2answers
158 views

Can someone show me a proof of the general solution for 2nd order homogenous linear differential equations?

I thought I had it figured out but there's a sort of 'leap of faith' at a pivotal point that annoys me. Can someone show me how to derive the general solution of an equation such as: ...
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0answers
152 views

Differential Equations with Deviating Argument

Is there literature available on solving differential equations of the type $$f(x,y(x),y(\kappa x),y'(x))=0,$$ where $\kappa$ is a given constant? I know about the book Introduction to the Theory and ...
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2answers
272 views

How to solve $\frac{dy}{dx}=5xy + \sin x$?

How do I solve $\frac{dy}{dx}=5xy + \sin x$ explicitly? With $y(0) = 1$. I am asked to use an integrating factor. What I did: $\frac{dy}{dx}-5xy = \sin x \\ \text{Integrating factor:} \ e^{\int{-5x\ ...
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1answer
122 views

Do I use Euler -method with Differentials correctly?

The book here (sorry not in English) on page 676: $$\begin{cases}y'=-y^{2} \\ y(0)=1\end{cases}$$ when $x_{k}=\frac{k}{5}$ which means $h=0.2$ i.e. $\Delta x=x_{k+1}-x_{k}=0.2:=h$. The task is ...
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1answer
318 views

Suggestions for a Global Analysis book

can somebody tell me some good books or lecture notes in "global analysis" ? I am a newcomer in this subject. thanks in advance. greetings trito
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1answer
195 views

A boundary value problem over an infinite interval

This is the edited version of the original problem, hopefully presented in a clearer manner. (I have also renamed this post with a more befitting title) Problem: $$y'(x) = ...
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2answers
233 views

$\frac{dx}{dt} + \alpha x = 1$ asymptotic behaviour and solution

I am trying to solve $\frac{dx}{dt} + \alpha x = 1$, $x(0) = 2$, $\alpha > 0$ where $\alpha$ is a constant. [some very badly done mathematics deleted] Continuing with Gerry's suggestion: ...
0
votes
1answer
49 views

Largest value in the functional $\int_0^\infty e^{-rt}( x^2+2x+\dot x^2)dt$?

I am trying to understand the second order linear differential equation and the answer here (Finnish) that I have translated below. Translation Problem What is the value of $x(t)$ where the ...
0
votes
0answers
38 views

Numerical Solvers to deal simultaneously with very different types of Oscillatory Behaviour

I am trying to solve these two related problems numerically: \begin{align} &f^{(\mbox{v})}(y) -(f^5 (y))'-\frac{1}{6}yf(y)=0\\ f'(0)=f'''(0)=0, &\quad f(y) \sim Cy^{(-1/7)}\exp(\gamma ...