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

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12
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1answer
179 views

Prove that there exists $t$ such that $0\le t\le T$ and $\int_0^Te^{-x}y'y''\,dx=\int_0^ty'y''\,dx$.

Let $y(x)$ be a solution to $y''+e^xy=0$. Prove that there exists $t$ such that $0\le t\le T$ and $$\int_0^Te^{-x} y'y'' \, dx=\int_0^ty'y''\,dx.$$
2
votes
3answers
93 views

How can I solve this differential equation?

How can I find a solution of the following differential equation: $$\frac{d^2y}{dx^2} =\exp(x^2+ x)$$ Thanks!
5
votes
2answers
314 views

The Green’s function of the boundary value problem

What is the Green’s function of the boundary value problem $$ \frac{\mathrm d^2 y}{\mathrm d x^2}-\frac{1}{x}\frac{\mathrm dy}{\mathrm dx}=1,\quad y(0)=y(1)=0, $$ this boundary problem is not self ...
4
votes
4answers
170 views

General solution of a differential equation

what is the general solution of this diff. equation $$x^2y''-4xy'+6y=x$$ Tried calling $y=xv$ but didnt work. ($x^2v''-2xv'+v=1$) what can I try else?
0
votes
1answer
35 views

Solution of $ y'' + NK\frac{y'}{t_f-t} + NK^2\frac{y}{(t_f-t)^2}=0 $

I have a question statement like this: Show that solution of $ y'' + NK\frac{y'}{t_f-t} + NK^2\frac{y}{(t_f-t)^2}=0 $ is $ y(t) = C_1(t-t_f) + C_2(t-t_f)^N $. N, K and tf are ...
1
vote
1answer
150 views

Matched Asymptotic Expansion - Stretching Transofrmation

I'm having problems getting my head around a stretching transformation in the method of matched asymptotic expansions. I'm reading Introduction to Perturbation Methods (by Holmes) and he discusses the ...
2
votes
3answers
2k views

Practical applications of first order exact ODE?

In elementary ODE textbooks, an early chapter is usually dedicated to first order equations. It is very common to see individual sections dedicated to separable equations, exact equations, and general ...
2
votes
2answers
1k views

A problem For the boundary value problem, $y''+\lambda y=0$, $y(-π)=y(π)$ , $y’(-π)=y’(π)$

For the boundary value problem, $y''+\lambda y=0$ $y(-π)=y(π)$ , $y’(-π)=y’(π)$ to each eigenvalue $\lambda$, there corresponds Only one eigenfunction Two eigenfunctions Two linearly ...
4
votes
1answer
130 views

On the differential equation $y''+y=0$

Consider the differential equation $$\frac{d^{2}y}{dx^{2}}+y=0$$ with initial conditions $y(0)=0$ and $y'(0)=1$. The solution is well known - $y=\sin(x)$. I know how to derive this solution, since the ...
1
vote
0answers
384 views

How to use Richardson extrapolation to derive modified Euler method?

For a given ODE $y'(t)=f(t,y)$, Euler's method is $$ y(t+h)=y(t)+hf(t,y(t)) + O(h^2) $$ It is said that by using Richardson extrapolation, we can improve it to $$ ...
3
votes
1answer
62 views

How to determine phase image

I need to sketch the phase image belonging to the following vector field (I'm sorry, I don't know the exact terms in English, so I have just freely translated them - thanks for sharing the correct ...
0
votes
1answer
108 views

Uniqueness Theorem for ODE

Let $g:[0, T]\times\mathbb{R}^n \to \mathbb{R}$, $a\in\mathbb{R}^n$. Consider the Cauchy problem $$\begin{cases}x'(t)=g(t,x) \quad a.e \quad s\in [0, T]\\ x(0)=a\end{cases}$$ where $g(t,x)$ is ...
1
vote
0answers
86 views

limit of a tricky multi-variable function

I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: ...
19
votes
1answer
433 views

How does one parameterize the surface formed by a *real paper* Möbius strip?

Here is a picture of a Möbius strip, made out of some thick green paper: I want to know either an explicit parametrization, or a description of a process to find the shape formed by this strip, as ...
0
votes
1answer
90 views

A basic question about partial differentiation

F is a smooth function of x and y, i.e. F(x,y). If $H(x)=F(x,0)$, when can I have $\dfrac{\partial F}{\partial x}(x,0)= \dfrac{dH} {dx}(x)?$ I think we can show this equality from the definition of ...
2
votes
0answers
129 views

The ordinary differential equation $\frac{d^2y}{dx^2}-q(x)y = 0$ , $0≤x<∞$ , $y(0)=1 $, $y'(0)=1$ multiple choice question

I am stuck on the following question: Assuming $$\frac{d^2y}{dx^2}-q(x)y = 0,\;\; 0 \le x \lt \infty ,\;\;y(0)=1,\;\;y'(0)=1$$ wherein $q(x)$ is monotonically increasing continuous function,then ...
1
vote
2answers
135 views

the Green function $G(x,t)$ of the boundary value problem $\frac{d^2y}{dx^2}-\frac{1}{x}\frac{dy}{dx} = 1$

the Green function $G(x,t)$ of the boundary value problem $\frac{d^2y}{dx^2}-\frac{1}{x}\frac{dy}{dx} = 1$ , $y(0)=y(1)=0$ is $G(x,t)= f_1(x,t)$ if $x≤t$ and $G(x,t)= f_2(x,t)$ if $t≤x$ where ...
6
votes
5answers
187 views

Check $\;y=\dfrac{\sin x}{x}\;$ is solution of $\;xy'+y=\cos x\;$

How can I check that $\;y=\dfrac{\sin x}{x}\;$ is a solution of $\;xy'+y=\cos x\;$?
1
vote
2answers
575 views

Clairaut's Equation Singular and General Solutions

I want to know how one how one would prove that the singular solutions to Clairaut's equation are tangent to the General solutions. so I have here: $$y(x) = xy' - e ^{y'}$$ Differentiating $$y' = y' ...
1
vote
1answer
65 views

A system of differential equations and its graph given an initial condition

For a system of differential equations: $$ x'=x^2-1,\\ y'=-y $$ (1)Draw the solution curve that starts at the intial point $Y(0)=(x(0),y(0))$. (2)For the above solution curve $Y(t)=(x(t), y(t))$, ...
1
vote
1answer
38 views

What type of differential equation is this?

$$ y''+3ty'+2y=3\sin(t) $$ So below is my description of the equation above, let me know whether they are correct or not: (1) it's linear (no square or higher power) (2) it's second order (obvious) ...
1
vote
1answer
49 views

A set where the operator of differentiation is the sum

Let X be the set of differentiable functions in real line. Find an infinity set $H$ in X such that $$ f,g\in H ~~\rightarrow (fg)'=f+g $$
0
votes
2answers
588 views

Omega limit set is invariant

In the ODE where $y'=f(y(t))$ and $y(0)=yo$. The omega limit set $w(yo)$ is positively invariant and also negatively invariant. I want to prove first that its positively invariant and then prove ...
4
votes
1answer
156 views

Solution of non linear ODE system is always positive if its initial valus is positive

Given a system of nonlinear differential equation \begin{eqnarray}\frac{dx}{dt}=2x(3-y) \\ \frac{dy}{dt}=3y(4-x)\end{eqnarray} If $r(t)=$($x(t)$,$y(t)$) is a solution of the system with initial value ...
0
votes
1answer
245 views

Harmonic Extension

Let be $u$ a harmonic function defined on an open set $\Omega \setminus \{p\} \subset \mathbb{C}$ of the complex plane. Show that if $u$ is bounded in a neighborhood of $p$ then $u$ admits a harmonic ...
1
vote
2answers
525 views

Nonhomogeneous second order ODE constant external force

I've been trying to solve this equation $$-\mu u'' + \beta u' = 1$$ where $u(0) = 0$, $u'(1) = 1$. So far the result I have is $$u = \frac{(exp(\frac{\beta x}{\mu})-1)\mu}{\beta ...
0
votes
1answer
47 views

Newton's rate equation, determine constant and re-use

I'm having trouble with the following homework question! A container is full of liquid at 185C and is cooled by another fluid which was maintained at a temperature of 15C. The temperature of the ...
8
votes
1answer
359 views

Solve this equation : $y'(x)+\frac{1}{x}=\frac{1}{y}$

General and particular solution for this first-order nonlinear ODE : $$y'(x)+\frac{1}{x}=\frac{1}{y}$$
2
votes
1answer
183 views

Change of variables in Belousov-Zhabotinsky reaction

Here I am referring to http://demonstrations.wolfram.com/TheBelousovZhabotinskyReaction/, but not only this. If you press "Download demonstration as CDF" (implying you have the necessary tools to open ...
0
votes
1answer
50 views

The property of ODE $y''+(1+\cos{x})y'+xy=\sin{x}$

$y''+(1+\cos{x})y'+xy=\sin{x}$ with $y(0)=2$. Is $y(0)$ the local maximum or minimum? What is the convexity near $x=0$ (in a region $(\epsilon,0)$ with $\epsilon$ small enough, is y(x) convex or ...
1
vote
2answers
62 views

Problem related to a differential equation

I am stuck with the following problem: Let $Y(x)=(y_{1}(x),y_{2}(x))$ and let $A$ is given by $$\begin{pmatrix} -3 &1 \\ k& -1 \end{pmatrix}.$$ Further, let $S$ be the set of values of $k$ ...
1
vote
1answer
160 views

Finding the particular solution

I was thinking about the problem that says: Suppose $y_{p}=x\cos(2x)$ is a particular solution of $y^{n}+\alpha y=-4\sin(2x),$ where by $y^{n}$, i mean $$ ...
0
votes
1answer
457 views

Write this piecewise function in terms of the unit step

\begin{align} f(t) = \begin{cases} 3t &\mbox{if } t \leq 3 \\ 12 & \mbox{if } 3<t\leq 7 \\ 0 & \mbox{if } t\geq 7 \end{cases} \end{align} I'm confused as to how I can write this in ...
0
votes
1answer
54 views

How to calculate the center manifold of the system of ODEs?

I've encountered a problem like this: \begin{eqnarray} \left\{ \begin{array}{l} \dot{y_1}=-y_1+y_2^2,\\ \dot{y_2}=y_2,\\ \dot{y_3}=y_1y_2 \end{array}\right.\notag \end{eqnarray} The linerized matrix ...
0
votes
1answer
114 views

Finding the eigenvalues for a $3\times 3$ matrix

With the matrix $A$ given by $$\left( \begin{array}{ccc} 0 & -1 & 0 \\ 0 & 1 & a \\ 1 & 0 & 1 \end{array} \right)$$ the solution to the initial value problem $x'=Ax$, $x(0) = ...
0
votes
2answers
70 views

How to solve $tx'' -3x'+x=0$, x(0) = 0

Is there a way to solve this problem using Laplace transforms when only one initial condition is given? I've gotten to a point in the problem where I have $$-s^2X'(s) +(1-5s)X(s) = x''(0)$$ Is ...
0
votes
2answers
113 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
3answers
105 views

Boundary value problem of a second order linear differential equation

A real-valued function $f$ defined on a closed interval $[a, b]$ has the properties that $f (a) = f (b) = 0$ and $f (x) = f'(x) + f''(x),\;\forall x \in [a, b]$. Prove: $$f (x) = 0, \;\forall x \in ...
2
votes
2answers
67 views

Solving an inhomogenous parameter dependent ODE

I was trying to solve the ODE \begin{equation} \ddot{r} r = \alpha(\dot{r}^2-1) \end{equation} where $\alpha$ is an arbitrary constant. There are some simple cases when $\alpha = -1 $ then you can ...
1
vote
1answer
130 views

Maple plot problem

I'm tryin to make a 3D plot of a differential-equation. I already found: $$\begin{align*} &x_1' = f(t) x_4 - x_2\\ &x_2' = -f(t) x_3 + x_1\\ &x_3' = f(t) x_2 + x_4\\ &x_4' = -f(t) x_1 ...
0
votes
1answer
46 views

Understanding Laplace transforms of equations multiplied by $t$

If I have an equation I want to solve, such as $tx''+t^2x'-3x=0$ using $x(0)=0$, how can I easily reason what the Laplace of the terms multiplied by $t$ would be? Can I do the following: ...
2
votes
2answers
115 views

Partial fractions: how can we tell if the numerator should be constant or linear?

For example, if I have $$f(s) = \frac{1}{s^2}\frac{1}{s+1}$$ How do I know that this can be rewritten as $$f(s) = \frac{As+B}{s^2}+\frac{C}{s+1}$$ as opposed to $$f(s) = ...
2
votes
2answers
172 views

Not empty omega limit set

Dynamical system is generated by: $x'=-x+f(x,y)$ $y'=-y+g(x,y)$ $f,g \in C^1$ and $f,g$ are bounded. Prove that the omega limit set of p: $\omega(p) \neq \emptyset$ for all $p \in \mathbb{R}^2$. ...
1
vote
2answers
217 views

How can we write a piecewise function in the form $u(t-a)f(t-a)$?

Given a piecewise function, such as $$f(t) = \begin{cases} 2, & \text{if }t \lt a \\ t^2, & \text{if }t \geq a \end{cases}$$ Or some other piecewise function, how can we write it in the form ...
1
vote
2answers
273 views

Use Laplace transform to find a solution for $tx''+x'+tx = 0$

I was hoping I could get some help checking my work through. $\mathcal{L}\{x'(t)\} = sX(s)-1$ and $\mathcal{L}\{x''(t)\} = s^2X(s)-s$ Using the equality $\mathcal{L}\{-tf(t)\} = F'(s)$ we can ...
1
vote
1answer
198 views

Differential equations: $f(x,y) dx + g(x,y) dy = 0$

$$(2xy + 3y^2)dx + (x^2 + 6xy - 2y)dy = 0$$ $$y(1) = -1/2$$ How do you solve this? I have just started learning Differential equations and I have some trouble. Is this equivalent with this? ...
1
vote
1answer
71 views

Simple Harmonic estimate

I know this is simple, and I see all the pieces of the puzzle are there, but I can't seem to get it. Let $u$ be a solution of $$\Delta u = f \;\;\; x \in B_4 $$ Then if we can bound $$\int_{B_4} ...
0
votes
1answer
48 views

Asymptotic behaviour of the solution to a certain ODE

Let $f:\mathbb{R} \rightarrow \mathbb{R}$, $f \in C^{1}$, and $f(0)=f(1)=0$, $f(x)>0$ for $x \in (0,1)$. Prove that solution (with maximal domain) $u$ of problem: $x'=f(x)$, $x(0)=x_{0} ...
2
votes
2answers
130 views

Self-Adjoint Operator

If I have a self-adjoint operator which operates on twice differentiable functions defined by: $$ Lx(t) = [k(t)x'(t)]' + g(t)x(t) $$ How can I show that $k(t)$ is real-valued given that $k \in ...
0
votes
1answer
90 views

Find the equation of the orthogonal family

If $\;\; \displaystyle {dy\over dt} = 2y(12-3y),\;$ and $y(0)=1$, what is the maximum value? Find the equation for the orthogonal family $y= Ce^{5x}.$