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

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5
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0answers
108 views

Uniqueness result in linear differential equation of degree $n$.

Suppose that $f$ is such that $$f^{(n)}=\sum_{j=0}^{n-1}a_jf^{(j)}$$ Some little work is needed to get to ($a_j=0$ if $j<0$) $${f^{(n + 1)}} = \sum\limits_{j = 0}^{n - 1} {\left( {{a_{j - 1}} + ...
5
votes
4answers
221 views

Special Differential Equation

I ended up with a differential equation that looks like this: $$\frac{d^2y}{dx^2} + \frac 1 x \frac{dy}{dx} - \frac{ay}{x^2} + \left(b -\frac c x - e x \right )y = 0.$$ I tried with Mathematica. But ...
5
votes
2answers
441 views

Regarding Ladder Operators and Quantum Harmonic Oscillators

When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator: Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
5
votes
4answers
335 views

Where can I find good, free resources on differential equations?

I'd like to know if there are any good online books, lecture notes, videos, tutorials, or similar that are free to the public (on differential equations). Suggestions are welcome!
4
votes
1answer
27 views

Existence and uniqueness of soluctions of $y'=xy^{2/3}$

It is asked to analyze the existance and uniqueness of solutions of the ode at every point $(x_o, y_o)$ $$y' = 3y^{2/3}$$ My attempt: We consider the initial condition $ y(x_o)=y_o$. If we consider ...
4
votes
3answers
91 views

Initial value problem for 2nd order ODE $y''+ 4y = 8x$

How can I go about solving this equation $y''+ 4y = 8x$? Progress I found the general solution for its homogeneous form. What I don't know is how to find its particular solution.
4
votes
2answers
102 views

Differential Equations with Discontinuous Forcing Functions

$$ y''+y'+1.25y = g(t), \quad t > 0, $$ $$y(0) = 0, \quad y'(0) = 0 $$ $$g(t) = \left\{ \begin{array}{ll} \sin{t} & 0 \le t < \pi \\ 0 & t \ge \pi \end{array}\right.$$ ...
4
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4answers
143 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
4
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0answers
73 views

Solving second order nonlinear ODE

Having the following second order ordinary differential equation: $$ \ddot{x} = a \cos(x) $$ where, $a$ is a constant. What's an approach to solve this kind of equation?
4
votes
1answer
119 views

Asymptotic estimate of an oscillatory differential equation

Let $f\in C^1(\mathbb{R} ,\mathbb{R} )$ and satisfying the condition: $$ \forall x >0, \quad f(x)>0, \forall x<0 , \quad f(x)<0 $$ Let $(\alpha, \beta) \in \mathbb{R^2}$. ...
4
votes
1answer
2k views

Solving an initial value ODE problem using fourier transform

I am a physics undergrad and studying some transform methods. The question is as follows: $y^{\prime \prime} - 2 y^{\prime}+y=\cos{x}\,\,\,\,y(0)=y^{\prime}(0)=0\,\,\, x>0$ I am having some ...
4
votes
1answer
154 views

Is Euler's lemma of fluid mechanics a nonlinear version of Liouville's theorem of ODEs?

Liouville's Theorem Consider the following linear system of ordinary differential equations: $$\tag{1} \dot{\mathbf{x}}=A(t)\mathbf{x}(t).$$ Let $\mathbf{x}_1, \mathbf{x}_2, \ldots, ...
4
votes
2answers
113 views

Time required to reach the goal when an object will be slowing down incrementally based on distance travelled?

I was thinking about this when flying on the plane which was approaching and slowing down. Assume an object is approaching its target which is at a certain initial distance d at time t0. It starts ...
4
votes
1answer
1k views

Convert Airy's equation, $y'' - xy = 0$, into Bessel equation, $t^2u'' + tu' + (t^2 - c^2)u = 0$

My professor has said that this will be an easy homework exercise. He suggested using change of variable $t = \dfrac{2}{3}x^{3/2}$, and then removing the first derivative term of the form $p(t) \dfrac ...
3
votes
2answers
111 views

Solving a particular system of differential equations

The problem I'm trying to solve is this: $X'(t) \in \mathbb{R}^3 \,, \, \omega = (\omega_1,\omega_2,\omega_3) $ Find the general solution for $$X'(t) = \omega \times X(t)$$ After doing the cross ...
3
votes
1answer
78 views

Analysis of stability of a linearized ODE with a periodic solution

I am asked to find the stability of the following ODE: \begin{equation*} \dot{y} = y^{2} + 2\cos(t)\sin(t) - \sin^{4}(t) \end{equation*} by linearizing around a particular solution $\eta = ...
3
votes
3answers
1k views

Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = ...
3
votes
2answers
74 views

Finding the general solution to a system of differential equations

How can I solve the following system of differential equations? I am getting confused with the constants of integration... $$\dot{x}=2x-(2+y)e^{y}$$ $$\dot{y}=-y$$ I know that $y=Ce^{-t}$ and the ...
3
votes
2answers
130 views

Liapunov Function for a Differential Equation

I have been trying to find a Liapunov function which would give me information about the stability of the following system of differential equations, however, I am not able to come up with any. The ...
3
votes
1answer
107 views

Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
3
votes
1answer
193 views

Definition of weak solution in $W^{1,2}(\Omega)$.

I have a problem: For $\Omega$ be a bounded domain in $\Bbb R^n$. We consider $$\left\{\begin{matrix} \Delta u-\lambda u =f \ \rm in \ \Omega & \\ u\mid_{\partial {\Omega}} =0 ...
3
votes
1answer
172 views

Small question about ODE

i have this question : Given three parameters $L,a$ et $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ > t\geq0$$ 1) Show that the ...
3
votes
3answers
574 views

Power Series Solution for $e^xy''+xy=0$

$$e^xy''+xy=0$$ How do I find the power series solution to this equation, or rather, how should I go about dealing with the $e^x$? Thanks!
3
votes
2answers
193 views

Green's function for $\frac{d^2}{dx^2} + \frac{1}{4}$

I am trying to solve the following differential equation $$y''+\frac{1}{4}y=\sin(2x)~~~y(0)=y(\pi)=0,$$ using Green's function. I have found the Green's function for the operator $y''+\frac{1}{4}$ to ...
3
votes
2answers
651 views

First-order nonlinear ordinary differential equation

How to solve this differential equation: $$x\frac{dy}{dx} = y + x\frac{e^x}{e^y}?$$ I tried to rearrange the equation to the form $f\left(\frac{y}{x}\right)$ but I couldn't thus I couldn't use $v = ...
3
votes
1answer
186 views

differential system on the torus

In a recent topic I've studied on complex analysis I had to study the differential system on the torus $\mathbb T^2:$ $$\begin{cases}\frac{\partial}{\partial y}u-\frac{\partial}{\partial ...
3
votes
2answers
257 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
75 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
56 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
65 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
4answers
201 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
133 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
279 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
754 views

How to reduce higher order linear ODE to a system of first order ODE?

Is there any general and systematic way of reducing the higher order linear ODE to a system of first order ODE? For example, assume we have $a_3x^{(3)}+a_2x^{(2)}+a_1x^{(1)}+a_0x=0$, then how do we ...
2
votes
1answer
124 views

resolve an non-homogeneous differential system

Solve the given non-homogeneous system: $$\dfrac{dx}{dt}=3x + 3 y - 2 z + e^t$$ $$\dfrac{dy}{dt} = x+y+ 2 z$$ $$\dfrac{dz}{dt}=x+3y+e^t$$ Can you help me understand a simple method to resolve the ...
2
votes
2answers
105 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
416 views

Stability of nonlinear system with borderline linearization

I have the following nonlinear system: \begin{align} ...
2
votes
3answers
116 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
925 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
777 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
568 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
582 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
168 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
104 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
445 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} ...