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

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2
votes
2answers
122 views

How to solve two point boundary value problem $ y'' + 2y = -x$

How to solve this differential equation $y'' + 2y = -x$ ? I started with $y(x)= c1 \cos(\sqrt(2)x) + c2 \sin(\sqrt(2)x)$, but i think i need to put some $Yp(x)$ for $-x$ inside the equation but I do ...
2
votes
1answer
42 views

Determining y(2) knowing that $y(1)=1+e^3$ and $ty'+(2t^2)y=2t^2$

How do I determine y(2) knowing that $y(1)=1+e^3$ and $ty'+(2t^2)y=2t^2$ ?
0
votes
1answer
70 views

How can I find this derivative

Suppose that $$\tau=F(\nu)$$ How do I check if F satisfies a differential equation $\frac{d\nu}{d\tau}$? Normally I could find nu in terms of tau, but F is extremly complicated. Here are my ...
3
votes
2answers
56 views

What is the integrating factor of $2xy'+x^{2}e^{1-x^2}y=2$

I know how to start solving it, its dividing everything by $2x$, but I can't solve $$\int \dfrac{x^2e^{1-x^2}}{2x}\,dx$$
0
votes
1answer
44 views

Scalar equation is uniformly asymptotically stable

Would you please help me to verify that if constant $a$ is positive, then the zero solution of the scalar equation $$ x'=-ax, $$ is uniformly asymptotically stable.
1
vote
3answers
58 views

Differential equation with an integral

I need to find function $f$ (the book doesn't specify whether $f(x)$ or $f(t)$) that satisfies the following equation: $$\int_0^xf(t)\,dt= {2 \over 3}xf(x)$$ Can anyone please tell me how this ...
1
vote
2answers
174 views

Stable/unstable equilibrium points

Consider: $$ \dfrac{dN}{dt} = -rN \left(1-\dfrac{N}{K_1}\right) \left(1- \dfrac{N}{K_2}\right), $$ where $r,K_1,K_2$ are constants s.t. $r>0 $ and $0 < K_1<K_2$. Find the the ...
0
votes
1answer
65 views

Physics problem - about a shell. Differential equation

Can you help me please, to write the differential equation for this problem, and give me an idea how to solve this equation. A shell of mass $2$ kg is shot upward with an initial velocity of $200$ ...
1
vote
2answers
147 views

Existence and Uniqueness of solution for an IVP

Consider the following IVP: $$ y'(x)=1+y^{2/3},\quad y(0)=0, $$ where the flux function is $f(x,y)=1+y^{2/3}$. According to Picard-Lindelöf Theorem, since $f_{y}$ is not continuous in any interval ...
1
vote
1answer
116 views

Green Function First Order DE

I'm having a bit of trouble working out the Green function for $$y' = f(x)$$ when the boundary condition is something like $y(a) = y_0 \neq 0$. I'll do it as though $y(a) = 0$ and at the end modify to ...
4
votes
2answers
262 views

how to solve $x^3y′′−xy′+y=0$

I tried to use Frobenius method to solve $$ x^{3}{\rm y}′′\left(x\right) − x\,{\rm y}′\left(x\right) + {\rm y}\left(x\right)=0, $$ but it does not work. And the solution most be $y_{1} = ax + b$. I ...
2
votes
1answer
74 views

Can anyone solve this ODE?

Is the following equation solvable analytically: $$uu''+au^3=b$$ Where $a,b$ are positive real numbers? As you can see, I used a u-sub to get to this equation, but I can't see any tricks. Also, DSolve ...
2
votes
1answer
184 views
2
votes
1answer
53 views

Why are Galerkin methods/FEMs used for solving PDEs and rather not ODEs?

I have not yet understood Galerkin methods and in general not the structural differences between ODEs and PDEs (of course I know the basics but not why PDEs ist so much different except that they ...
1
vote
0answers
39 views

The Laplace transform - does it have an associated differential operator, if the kernel is to be viewed as a Green's function?

I've begun learning about Green's functions, and if I understand correctly, the Green's function for a linear differential operator $L$ with appropriate boundary conditions is the kernel for the ...
1
vote
1answer
102 views

Good ODE Books That Explain How Solution Methods Came To Be and Their Justifications

As part of the mathematics program offered at my college, I took an introductory ODE course a few semesters back. This was the one math course in my entire college career that I was totally lost in. ...
0
votes
1answer
199 views

For a 2nd order linear ODE,find the interval where some non-trivial solutions remain bounded and some become unbounded

$$y''+(2\alpha-3)y'+\alpha(\alpha-3)y = 0$$ Determine all values of $\alpha$, if any, for which some non-trivial solutions remain bounded and some become unbounded as $t \rightarrow \infty$. The ...
11
votes
0answers
209 views

Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} ...
1
vote
0answers
26 views

Vectors evolving by $\frac{\partial \vec{x}}{\partial t}=\vec{U}(\vec{x})$: maximize effect of perturbation.

Suppose we have a finite dimensional real vector space $V$ equipped with a norm $\|\cdot\|$ which is given by $$\|\vec{x}\|^2 = \vec{x}^TX\vec{x},$$ where $X$ is a matrix and $x\in V$ is in ...
1
vote
0answers
104 views

How to solve analytically or simplify this coupled system of ODEs?

I have a coupled system of ODEs: $$\cases{ i\frac{\text{d}y_1}{\text{d}t}=A f(t)y_2(t)+E_1 y_1(t)\\ i\frac{\text{d}y_2}{\text{d}t}=A f(t)y_1(t)+E_2 y_2(t) }\tag1$$ Here $f(t)$ is a periodic function ...
1
vote
0answers
126 views

Differential equation with random variable

How can I derive analytically or compute numerically the solution to following differential equation $$ dy/dt = y\cdot X\cdot (y\cdot X - g(y,X))\cdot X $$ where X is a random variable (e.g. from a ...
7
votes
6answers
1k views

Differentials Definition

Please define differentials rigorously such that they give a consistency to their use in the following links. I have read Is $dy/dx$ not a ratio? What is the practical difference between a ...
0
votes
4answers
442 views

Ansatz of particular solution, 2nd order ODE

Find the particular solution of $y'' -4y' +4y = e^{x}$ Helping a student with single variable calculus but perhaps I need some brushing up myself. I suggested y should have the form $Ce^{x}$. This ...
1
vote
1answer
60 views

Lotka-Volterra equations mistake

I have a problem with the Lotka-Volterra equations themselves. I believe that they might be wrong. Here is my reasoning - I would appreciate it if someone could find a flaw in it! The equations are ...
2
votes
4answers
2k views

Solving $y'' - y = 0$

I am attemtping to solve $y'' - y = 0$ I come to this solution, by using something like $\frac{dy}{dx} = p$ So it does $\frac{dp}{dy} \cdot \frac{dy}{dx} - y = 0$ Which gives $\frac{dp}{dy} \cdot p ...
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 ...
0
votes
0answers
26 views

Existence and uniqueness of solutions for the following cauchy problem

Let $ f \in C^n([-a,a])$ and $x \in [-a,a]$. Consider the following Cauchy problem: $ \begin{cases} x' = | f(x) - f(x_0) - f'(x_0)(x-x_0) - \dots - \dfrac{f^{(n)}(x_0)}{n!}(x-x_0)^n | \\ x(0) = x_0 ...
0
votes
0answers
61 views

In the Proof for Existence of an Unique Solution (for Differential Equations), why $\frac{\partial f}{\partial y}$?

Theorem: Let $R$ be a rectangular region in the $xy$-plane defined by $[a,b]\times[c,d]$ that contains the point $(x_0,y_0)$ in its interior. If $f(x,y)$ and $\dfrac{\partial f}{\partial y}$ are ...
0
votes
2answers
42 views

A system of differential equations involving limits

Let $u(t)=(u_1(t),u_2(t))$ be the unique solution of the problem: $\frac{d}{dx}(u(t))=Au(t)$, $t\gt0$ $u(0)=u_0$ where $u_{0}=(1,1)$ and $A$ is a symmetric $2$ into $2$ matrix such that $tr(A)\lt0$ ...
0
votes
1answer
65 views

Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
3
votes
3answers
110 views

Solution to Hamilton-Jacobi differential equations

Let $H(x,y)$ be a $C^2$ function on $\mathbb{R}^2$ and let $(x(t),y(t))$ be a solution of the Hamilton-Jacobi equations $$\frac{dx}{dt}=\frac{\partial}{\partial ...
0
votes
1answer
78 views

Inverse Fourier Transform of the output, Y(f)

A linear system is defined by the differential equation: $$ y''(t) + 4y'(t) + 25y(t)= x(t) $$ The transfer function of this system is: $$ H(f) = \frac{Y(f)}{X(f)}= \frac{1}{(2\pi fj)^{2}+ 4(2\pi ...
0
votes
1answer
111 views

How to find a smooth solution - initial value problem

Solve the initial value problem: y''+4y=f(t) y(0)=1, y'(0)=0 f(t) = 1-t on [0,1] t-1 on [1,2] And f(x) is a periodic function, and f (t+2) =f (t) for all t So I solve it for [0,1] and for [1,2] ...
2
votes
1answer
294 views

$\omega$ and $\alpha$ - limit sets

I'm reading a book that defines the $\omega$ and $\alpha$ - limit sets of a differential equation respectively as : $\alpha (x)$ = $\cap_{y \in \gamma (x)}$ $\overline \gamma^{-} (y)$ $\omega (x)$ ...
-1
votes
1answer
270 views

Solve IVP using Laplace transform?

Solve the IVP using Laplace transform: $$y'' + 4y = g(t); \hspace{5 pt}y(0) = 1, \hspace{5 pt} y'(0) = 3$$ and $$g(t) = 3 sin (t), 0 \leq t < 2\pi; \hspace{10 pt} 0, 2\pi \leq t$$ Take step ...
0
votes
2answers
74 views

Eigenvalues of $\frac{d^2}{dx^2}$ in $C^2(\mathbb{R})$

Consider the eigenvalue problem \begin{equation} \left\{ \begin{array}{l} \Phi \in C^{2}(\mathbb{R}) \ \text{and bounded }\\ -\Phi^{''}(x)=\lambda\Phi(x), \ x\in \mathbb{R}. \end{array} \right. ...
0
votes
1answer
62 views

stationary function of an integral

Find the stationary function $y=y(x)$ of the integral $\int_o^4[xy'-(y')^2]dx$ satisfying the conditions $y(0)=0$ and $y(4)=3$. I don't know what a stationary function is. Can you anyone suggest me ...
0
votes
1answer
80 views

map, stability of the fixed point, cobweb

Map => $x_{n+1}=\sin x_n$ Show the stability of the fixed point $x^*=0$ is not determined by the linearizatin. Using the cobweb to show $x^*=0$ is stable. I took derivative of $\sin x_n$ and put ...
2
votes
2answers
68 views

By finding solutions as power series in $x$ solve $4xy''+2(1-x)y'-y=0 .$

By finding solutions as power series in $x$ solve $$4xy''+2(1-x)y'-y=0 .$$ What I did is the following. First I let the solution $y$ be equal to $$y =\sum_{i=0}^{\infty} b_ix^i =b_0 ...
0
votes
3answers
394 views

solve the initial value problem ,by Taylor's method of order $N=3$

solve the initial value problem ,by Taylor's method of order $N=3$ $y'(t)=ty(t)+(1-t)e^t,0\le t\le 2,y(0)=1$ with an accuracy of $5 \times10^{-3}$ first we consider the taylor expansion of $e^x$ $ ...
0
votes
1answer
72 views

How to form a differential equation, given temperature and direction of heat flow

I am given the following information: $$T(x, y) = xy − x$$ where $T$ represents temperature. Heat flows in the direction $−\nabla T$ (perpendicular to the isothermals). How do I use this to make a ...
1
vote
1answer
135 views

Find the index of the equilibrium points of the system (Question on solution)

I have the following system: $$\dot{x} = 2xy$$ $$\dot{y} = 3x^2-y^2$$ I have the following solution: The system has one equilibrium point at the origin. Let the curve $\Gamma$ surrounding the origin ...
0
votes
1answer
77 views

Question on dynamical system

i have this exercise : we consider the following model : $$ \begin{cases} x'& = x(4-x-y)\\ y'&=y(2+2\alpha-y-\alpha x) \end{cases} $$ a) Find the critical point $P$ does not depend on ...
0
votes
2answers
121 views

Existence of invariant set in dynamical system generated by ODE

Is there any nonempty, compact and invariant set in dynamical system generated by this system of equations? $x'=x+\sin{(xy+2)}-7$ $y'=-y+\arctan{(x^2+y^3-6)}$ My idea is to use this fact: Not ...
2
votes
2answers
163 views

Unstable fixed point

Consider the system $\dot{x} = x(1-4x^2-y^2)-\frac{1}{2}y(1+x) $ $\dot{y} = y(1-4x^2-y^2)-2x(1+x) $ Show that origin is an unstable fixed point I made $\dot{x} = 0$ and $\dot{y}=0$ and $\dot{x} = ...
2
votes
2answers
102 views

finding the potential v(x,y)

Consider the system $\dot{x}=3x^2-1-e^{2y}, \dot{y}=-2xe^{2y}$ 1)Show that $\frac {\partial{f}}{\partial{y}}=\frac {\partial{g}}{\partial{x}}$ 2)Find the potential $V(x,y)$ 3)Show trajectories ...
0
votes
1answer
26 views

Reducing PDE's to canonical form - does order of differentiation matter?

Silly question, but when you're reducing PDE's to canonical form, does order of differentiation matter? I think I remember (vaguely) from calculus that d^2/dxdy = d^2/dydx - does this also apply to ...
2
votes
1answer
66 views

Derive a perturbation of period $2\pi$, to order $\epsilon$

I have the following problem: In the equation $\ddot{x}+\Omega^2x+\epsilon f(x) = \Gamma \cos t$, $\Omega$ is not close to an odd integer, and $f(x)$ is an odd function of x, with expansion, $$f(a\cos ...
0
votes
1answer
200 views

Time-delay differential-difference equation

Is it possible that the system $$ \begin{cases} 2\dot{q}(t) + \dot{q}(t-1) + \dot{q}(t+1) = k & \text{if} \hspace{5mm} 0 \leqslant t \leqslant 2 \\ \dot{q}(t) + \dot{q}(t-1) = c & \text{if} ...
1
vote
1answer
84 views

Bounded second derivative of solution to first order differential equations

I am studying numeric solutions of differential equations, and part of my reading is found in Simmonds' book, Differential Equations with Applications and Historical Notes. Although the chapter on ...