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

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0
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18 views

Express $y$ in terms of $x$

After solving some differential equation I arrived at $$2ln(y-1) +(y-1)=(x-3)+3 ln(x-3)+c$$ but I can't write $y$ in terms of $x$ to find explicit solution.
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0answers
8 views

Green function integration

When I'm trying to find the Green Function of Helmholtz equation for a cube $0≤x,y,z≤L$ $$\nabla^2u+k^2u=\delta(\vec{x}-\vec{x}')$$ where u=0 on the surface. I set to find the green function where ...
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0answers
6 views

Find a vector field $\mathbb{Y}$ satisfying $L_{\mathbb{X}}\mathbb{Y}=\mathbb{Z}$

Let $\mathbb{X}$ be the vector field on $\mathbb{R}^2$ given by $\mathbb{X}=(1,y)$. Let $\mathbb{Z}$ be the vector field on $\mathbb{R}^2$ given by $\displaystyle \mathbb{Z}(x,y)= \bigg( ...
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0answers
12 views

A question about fundamental matrix of periodic system $x'=A(t)x$

$X(t)$ is a fundamental matrix of linear differential equation $x'=A(t)x$ where $A(t)$ is a periodic matrix with period $T$ . Show that there exist a non-singular matrix like $C$ such that for ...
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0answers
13 views

A-stability of Runge-Kutta methods

I am studying Runge-Kutta methods, but I can't understand why explicit Runge-Kutta methods are not A-stable. Someone can explain it to me?
2
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0answers
26 views

Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{x\to\infty} ...
0
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0answers
6 views

Find a function to satisfy a necessary condition on a system of pdes

Consider the following set of PDE's $\displaystyle \frac{\partial u}{\partial x}(x,y)=f(x,y,u(x,y))$ $\displaystyle \frac{\partial u}{\partial y}(x,y)=1$ $u(x_0,y_0)=u_0$ Show ...
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3answers
56 views

How to solve a differential equation?

I'm trying to solve the system $$\frac{d^4x}{dt}+4x=0\quad ,\quad\frac{d^3x}{dt}+x=0$$. However, I don't know of any method of tackling such a problem. Can anyone please provide a route to a solution? ...
0
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3answers
35 views

Question on matrix exponential

Let $A$ be a real matrix with real eigenvalues $\lambda_k$ and complex eigenvalues $\alpha_ k \pm i\omega_ k$ , all of which are simple. I'm trying to show that every element of the matrix $e^ {tA}$ ...
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1answer
18 views

functions U and L solution of a differential equation

Solving this differential equation with an online calculator: $$-(a z+b) y+(c z+d) y''+cy' = 0$$ I obtain something like: $$y(z)=C_1 \exp\left(\frac{-\sqrt{a}z}{\sqrt{c}}\right) ...
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0answers
7 views

the equation for transverse vibration of a string

How can we derive the equation for transverse vibration of a string in a medium of which the resistance is proportional to the first power of the velocity We have to achieve the equation ...
0
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1answer
34 views

Solution to h'(t) = h(t)^j : Wolfram Alpha mistake?

This should be a quick and easy one. Trying to teach myself a bit more about differential equations, so I put the following equation into Wolfram Alpha: $$h'(t) = h(t)^j$$ It gave me the following ...
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2answers
28 views

Show $u\in H^1(B(0;1/2))$ is holder continuous, where $u$ is a weak solution to $-\Delta u+cu=f$ for some $c\in L^q$ for some $3/2<q<2,$.

If $u\in H^1(B)$, $B=\lbrace x\in\mathbb{R}^3, |x|<1/2\rbrace$ is a weak solution to $$-\Delta u+cu=f$$ for some $c\in L^q$ for some $3/2<q<2,$ and $f\in C^\infty$, then show $u$ is holder ...
2
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1answer
27 views

How to prove that solution of ODE is even function?

Could you please give me some hint how to prove this statement: If $f(x)$ is solution of $y'=4x^3e^{-|y|}$ then $f(x)$ is even function. It is obvious that $f(x)$ increasing for all $x>0$ and ...
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1answer
16 views

Unable to solve particular solution for non homogeneous second order diff. eq.

The book I'm following jumps many "obvious" steps and sometimes I can't follow up. I have the following non homogenous equation. However I'm unable to find the particular solution since I have so many ...
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1answer
18 views

Laplace transform of a differential equation??

Find unique solution of $y′′ + y = f$ using $y(0) = y′(0) = 0$ and periodic function $f(t) = t$ if $0 \leq t < 2\pi$ Attempted work: $L[y'' + y ] = L[f(t)]$ $L[y''] + L[y] = L[f(t)]$ $s^2 L[y] ...
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1answer
16 views

Variant of Picard-Lindelof theorem

Question Let $I=[0,a]$ and define the norm $||f||_{\lambda}=\sup_I |e^{-\lambda x}f(x)|$ for $f\in C(I)$. Let $\phi:\;\mathbb{R}^2\to\mathbb{R}$ satify $|\phi(x,u)-\phi(y,v)|\leq\rho |u-v|$ for all ...
0
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1answer
17 views

Finding complete general solution of differential equation with repeated roots (undetermined coefficents)

How do you get a complete general solution for a differential like this? $y^{\prime\prime}+6y^{\prime}+9y=14e^{-3x}$ This is what I have so far for the first part of the problem: $yp=Ce^{-3x}, ...
2
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1answer
23 views

Exponential of Matirx

So, I'm wondering if there is an easy way (as in not calculating the eigenvalues, Jordan canonical form, change of basis matrix, etc) to calculate this exponential e^At with A (0 9) (-1 0) I'd ...
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1answer
22 views

I want to find Euler-Lagrange equation for the given functional.

I want to find Euler-Lagrange equation for the following: $$J(u) = \int \left( \frac{\psi(x) u + \dot{u}}{\psi(x)u - \dot{u}} \right)dx, \text{where} \ \psi(x) \ \text{is an explicit function of} \ ...
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0answers
34 views

calculate the second derivative using `ode45`

I have a second order differential equation. I am using ode45 to solve the problem. ode45 converts the equations to the first ...
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1answer
30 views

why are these two power series the same

$$-\sum_{\color{red}{n=1}}^{\infty}nc_{n}x^{n}=-\sum_{\color{red}{n=0}}^{\infty}nc_{n}x^{n}$$ How come one starts at $1$ and the other starts at $0$ yet their equal? Do they both equal infinity?
3
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0answers
24 views

Solving PDE by Laplace Transform

Use Laplace transforms to solve the boundary value problem $$Y_{xx}(t,x)-2Y_{tx}(t,x)+Y_{tt}(t,x)=0, \quad 0<x<1, t>0$$ $$Y(0,x)=Y_t(0,x)=0, \quad 0<x<1$$ $$Y(t,0)=0, \ Y(t,1)=F(t), ...
0
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1answer
26 views

What type of differential equation is that?

Good day. I can't understand what type this DE has $ (7x-8y)y'=2x^2-y $ I guess it can't be homogeneous or separable equation. And it seems what it is not a linear equation.. Maybe it is Bernoulli ...
3
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1answer
24 views

Solving a system of Differential Equations: arbitrary constants

For a research project I am carrying out I am required to solve the system: $\frac{dp}{dt} = -lp $, $ \frac{dc}{dt} = lp - kc $ with initial conditions $p(0) = p_0 $ and $c(0) = 0 $. Here, $p,c$ ...
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 ...
0
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1answer
27 views

Laplace transform of a differential equation?

Find the unique solution of $y''+ y = f$, $y(0) = y'(0) = 0$ with the $2\pi$ periodic function given by $f(t)=2\pi \sin(t)$. I am having trouble setting up and starting the the question. I would be ...
0
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1answer
11 views

Differential equations resonance

I've got the question 'Solve (c^2)y ' ' + y = 0, y(1)=1, y'(0)=0. Show that as c->0, the solution does not tend to a limit'. From solving the equation I got the roots as +-(1/c)i, and then using set ...
1
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2answers
30 views

When do you drop the absolute value from ln|x| + C when integrating $\frac{1}{u}du$

Given: p(t) represents the number of cats, when t>=0. Given: p(t) is increasing at a rate directly proportional to $800-p(t)$ So, I represent this as: $\frac{dp}{dt}= k(800-P)$ I want p(t), so I ...
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0answers
24 views

Asymptotic behavior of the solution of a 2nd order linear ordinary differential equation

In studying the harmonic oscillator, we encounter the equation $$ f'' +(2E - x^2) f = 0$$ What is the asymptotic behaviour of the solution to this equation for a generic $E$? Any good book on ...
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2answers
28 views

solvability condition for differential operator

While reading the research article I came across following derivation, given a self-adjoint operator, \begin{eqnarray} L = \frac{d^2}{dx^2} + f(x) \end{eqnarray} \begin{eqnarray} L\psi_1(x) ...
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1answer
21 views

Uniqueness of differential equation solutions

I need to solve this DE $$y'' - 2x^{-1}y' + 2x^{-2}y = x \sin x \tag{*}$$ I found the complementary functions to be $x^2$ and $x$, and also noticed by guessing that the particular integral is $y = - ...
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0answers
14 views

Differential Equation and dynamical system [on hold]

How to plot the direction field from a autonomous system using mathematica.write command also.
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0answers
26 views

necessary and sufficient conditions in ODE theory [on hold]

I have trouble writing proofs when studying the abstract theory of ODE. For instance, I have trouble proving the existence of some special solutions of a given system of nonlinear ODE. In particular, ...
4
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1answer
42 views

General solution to $f^{(n)}=f$ but $f^{(k)}\ne f$ for $k<n$

We know that $$\frac{d}{dx}e^x=e^x$$ and $$\frac{d^4}{dx^4}\sin(x)=\sin(x)$$ What is the general solution $f$ to $$\begin{equation} \begin{split} \frac{d^n}{dx^n}f(x)&=f(x) \\ ...
2
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1answer
20 views

Growth of plant in greenhouse

The following problem came up in an exam I sat recently. I got 113cm, but I'm quite unsure about my method. Is someone able to go through the working and explain the problem? Of course, I don't ...
3
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1answer
40 views

If a differential operator $C$ factors as $AB$, then every solution of $C(y)=0$ has the form $y=y_1+y_2$ with $A(y_1)=0$ and $B(y_2)=0$

Given two constant-coefficient operators $A$ and $B$ whose characteristic polynomials have no zeros in common. Let $C = A B$. First part of question is Prove that every solution of the ...
0
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1answer
15 views

Simple problem about Laplace Equation in a domain

Suppose that "$u$", is solution of the problem $$\triangle u=0, r<R $$ $$u_{r}(R,\phi)=f(\phi), 0<\phi\ < 2 \pi$$ Show that $$\int_{0}^{2 \pi}{f(\phi)d\phi}=0$$ I know what this question ...
3
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2answers
26 views

Show stable node or spiral cannot occur

If I have the equation: $$\ddot{x} + f(\dot{x}) + g(x) = 0$$ where $f$ is even and $f$ and $g$ are both smooth, how do I show that the equilibrium points cannot be stable nodes or spirals? What I've ...
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1answer
21 views

Systems of First Order Linear Equations, finding P(t) from two given vectors

Consider the vectors $x^{(1)}(t) = (t,1)$ and $x^{(2)}(t) = (t^2, 2t)$ I computed the Wronskian which is t^2. I also know that it's continuous everywhere except when t=0. But I was wondering how to ...
1
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1answer
41 views

How do first integrals help you solve differential equations?

I am reading about Euler-Lagrange equations and this particular section is a little unclear. Consider the differential equation $$\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix} = \begin{bmatrix} ...
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0answers
23 views

Find the eigenvalue and eigenfunction of the boundary value problem

By setting $y=\frac{u}{\sqrt{x}}$, find the Eigenvalues and Eigenfunction for a boundary value problem: $$y'' + \frac{y'}{x} +\Big(λ- \frac{1}{4x^2}\Big)y = 0 ,\ \ y(\pi)=y(2\pi)=0$$ The only ...
3
votes
2answers
43 views

Why is $f(x) = x + \frac{1}{x}$ a mapping contraction?

Why is $f(x) = x + \frac{1}{x}$ a mapping contraction? The metric space in question is $[1,\infty)$. Also, if this were a contraction, wouldn't it have a fixed point by Banach's theorem? It looks to ...
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0answers
15 views

Differential equation and integration approximation magic

Say we have a differential equation: $$df(\mu) = g(f(\mu))dv(\mu)$$ I was wondering under what conditions we get something like this (integrating from $\mu_1$ to $\mu_2$): $$\int df(\mu) \approx ...
5
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2answers
32 views

Differential Equations Constant

The function $y(x)$ satisfies the linear equation $$y'' + p(x)y' + q(x)y = 0.$$ The Wronskian $W(x)$ of two independent solutions, denoted $y_1(x)$ and $y_2(x)$, is defined to be $$W(x) = ...
0
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0answers
17 views

How to reduce order of this ODE

I want to reduce this ODE to lower order but I am confused in some steps. Can someone comment? $$ AB\frac{d^3u}{dz^3}+C(D-z)\frac{du}{dz}=0, \,\, 0<z<L $$ $A,B,C,D,L$ are constants, all ...
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0answers
25 views

Find an integrating factor such that $y'=\frac{1-x+y}{x-y}$ is exact

Yet another question of this sort, and hopefully the last. In the previous question I posted, we were lucky enough and the integrating factor was a function of only one variable, the ansatz $\mu_y=0$ ...
0
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1answer
40 views

What does “two polynomials have no zeros in common” mean?

The question is Given two constant-coefficient operators $A$ and $B$ whose characteristic polynomials have no zeros in common. Let $C=AB$... What does that mean by "no zeros in common"?
4
votes
2answers
28 views

Ordinary differential equations of the form $M(x,y)dx+N(x,y)dy=0$ question

An ODE of the form $M(x,y)dx+N(x,y)dy=0$ is called "good" if $\frac{\partial (M(x,y))}{\partial y}=\frac{\partial (N(x,y))}{\partial x}$ We are given the differential equation ...
0
votes
1answer
15 views

If $F(t,x)$ decreases in $x$ for every $t$, show that if $f,g$ satisfy the equation $x' = F(t,x)$, then $|f(t)-g(t)|$ monotonically decreases.

Given a decreasing function $F(t,x)$ by $x$ for every $t$, show that if $f,g$ satisfy the equation $x' = F(t,x)$, $|f(t)-g(t)|$ monotonically decreases. I've tried deriving, I've tried plugging in ...