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

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963 views

Solving $-u''(x) = \delta(x)$

A question asks us to solve the differential equation $-u''(x) = \delta(x)$ with boundary conditions $u(-2) = 0$ and $u(3) = 0$ where $\delta(x)$ is the Dirac delta function. But inside ...
7
votes
3answers
298 views

Find solutions of the differential equation $3x^2y''+5xy'+3xy=0$.

Find all the solutions of the form $y(x)= x^m \sum_{n=0}^{\infty} a_nx^n, \ x>0 (m \in \mathbb{R})$ of the differential equation $3x^2y''+5xy'+3xy=0$. That's what I have tried: Since $x>0$ the ...
7
votes
3answers
554 views

Common term for differential equations and recurrence relations

Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you ...
7
votes
3answers
300 views

Why it is absolutely mistaken to cancel out differentials?

In many of my physics courses, (don't worry, this is a mathematics question!) My teachers cancel out differentials, and every time, they say: "If a mathematician saw me canceling out this ...
7
votes
2answers
843 views

Manifold interpretation of Navier-Stokes equations

I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines ...
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5answers
2k views

Modelling forces acting on a sail

I'd like to create a basic model of the forces acting on a sail (wind sail, like a tail ship) A couple of things I was thinking about: 1) Can create a very simple model where wind is 'one' force ...
7
votes
3answers
402 views

Finding matrix exponential

I am trying to compute the matrix exponential for $$A=\left( \begin{array}{ccc} 1 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 1 & -1 ...
7
votes
2answers
414 views

solving the integral $ \int_0^\infty \dfrac{\sin xt}{x(x^2+1)} dx $

How to solve the following integral by differential equations techniques? $$ \int_0^\infty \dfrac{\sin xt}{x(x^2+1)} dx $$
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3answers
928 views

Square root of differential operator

If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$. How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number) For ...
7
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3answers
356 views

Counterintuitive PDE

After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following: Given the PDE $\dfrac{\partial f}{\partial x} = 0 \quad $ on $U ...
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3answers
954 views

Solve $\frac{dx}{dt} = x^3 + x$ for $x$

This is a seemingly simple first order separable differential equation that I'm getting stuck on. This is what I have so far: $$\frac{dx}{dt} = x^3+x$$ goes to $$\frac{dx}{x(1+x^2)} = dt$$ Now ...
7
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1answer
1k views

How can I show that $y'=\sqrt{|y|}$ has infinitely many solutions?

Show that the first order differential equation $y'(x)=\sqrt{|y(x)|}$ with intial value $y(1/2)= 1/16$ has infinitely many solutions on the interval [−1, 1]. My thought were to show that this ...
7
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3answers
688 views

How can I solve the differential equation $y'+y^{2}=f(x)$?

$$y'+y^{2}=f(x)$$ I know how to find endless series solution via endless integral or endless derivatives and power series solution if we know $f(x)$. I also know how to find general solution if we ...
7
votes
3answers
2k views

Why does the absolute value disappear when taking $e^{\ln|x|}$

I have noticed that if you have an equation (after integrating) such as $$\ln|y| = \ln|x| + c,$$ and you further simplify it using the law of exponents, you get $$e^{\ln|y|} = e^{\ln|x|+с},$$ which is ...
7
votes
2answers
224 views

Estimating rate of blow up of an ODE

Suppose I have a differential equation $x'=f(x)$ and $f(x)>0$ grows super-linearly. I.e., $\lim_{|x| \rightarrow \infty} |f(x)|/|x| \rightarrow \infty$. Several related questions: (1) Can I ...
7
votes
1answer
227 views

Power series $x f''(x) + f'(x) + xf(x) = 0$

Find a power series with radius of convergence $R = \infty$ such that $$f(x) = \sum_{n=1}^{\infty} a_{n}x^{n}$$ satisfies $$x f''(x) + f'(x) + xf(x)= 0, \forall \mbox{ } x \in \mathbb R.$$ How ...
7
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4answers
436 views

Show $f''+vf' +\alpha^2 f(1-f)=0$ has solutions satisfying $\lim_{x \to - \infty}f=0$ and $\lim_{x \to \infty}f=1$ given $v\leq -2\alpha < 0$

I posted this question before but I took a completely different approach here, that's why I reposted as my previous question was already very long and took a different approach from here. I am given ...
7
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3answers
219 views

How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$?

How do I show there are no elementary function solutions for the differential equation $f''(x)=f(\sqrt{x}), x>0$ in the $C^2(0,\infty)$ space solutions?
7
votes
2answers
247 views

Solutions to the equation $y^{(n)} y = 1$ for even $n$

A long time ago I was curious about the closed-form solutions to the equation: \begin{equation*} \frac{d^{n}y}{dx^n} y = 1. \end{equation*} For $n$ an odd number, try $y = A x^k$. Then $y^{(n)} = A ...
7
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1answer
177 views

Nonlinear 1st order ODE involving a rational function

$$y'=\frac{-6x+y-3}{2x-y-1}$$ Is there a foolproof method for tackling equations of the form $y'=\dfrac{ax+by+c}{dx+ey+f}$ ? I've tried a few substitutions (like $y=vx$ and $v=2x-y-1$, neither of ...
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2answers
401 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem. Let $f(x):\Omega\longrightarrow \mathbb R^n$ be a $C^2$ function where $\Omega$ is an open subset of ...
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5answers
279 views

Solving $P(x,y)dx + Q(x,y)dy =0$: interpretation in terms of forms

I asked a similar question here which I will formulate more sharply: When we write a differential equation as $P(x,y)dx + Q(x,y)dy = 0$, what is the interpretation in terms of differential forms? ...
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1answer
195 views

prove that the following function is: $f(x) = 0$

let $f: [0,1] \to \mathbb R$ , $f$ is differentiable $f(0) = 0$ $|f'(x)|\le|f(x)|$ for $x\in [0,1]$ prove that $f(x) =0$ for $x\in [0,1]$ i believe that i need to somehow use the ...
7
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1answer
176 views

solution of $y' = \exp \left(-\frac yx\right) + \frac yx$

Could you help me to solve equation $$y' = \exp \left(-\frac yx\right) + \frac yx;\quad y(e) = 0$$ I know how to solve 1st order linear de like $y' = \exp \bigl(-\frac 1x\bigr) + \frac yx$ but here ...
7
votes
2answers
326 views

Is the Laplacian surjective on $C_0^{\infty}$?

Let $M := C_0^{\infty}(\mathbb{R}^n)$ denote the smooth maps with compact support. Then we have a map $\Delta:M\rightarrow M,\,\, f\mapsto \Delta f$, where $\Delta f = \sum_{i=1}^{n} ...
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2answers
3k views

Exponential of the differential operator

I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$ (ref. ...
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1answer
234 views

Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
7
votes
2answers
95 views

Is there differentiable $f: {\mathbb R}_+ \to {\mathbb R}_+$ such that $f'(x) > f(x)^2$ for all $x$?

Is there positive differentiable $f: {\mathbb R}_+ \to {\mathbb R}_+$ such that $f'(x) > f(x)^2$ for all $x$? It seems like the answer is no because such a function should have a vertical asymptote ...
7
votes
2answers
594 views

Sturm Liouville with periodic boundary conditions

Background and motivation: I'm given the boundary value problem: $$y''(x)+2y(x)=-f(x)$$ subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$. EDIT: These were not given to be zero !! Maybe this ...
7
votes
1answer
507 views

Sloppy notation for differential equations

Why does one often use the following notation for differential equation: $$ y'=f(t)y$$ (this is just a particular example) ? What bothers me with this notation, which I have encountered in ...
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3answers
616 views

A simple question about sine and cosine

I have been thinking about all of the different ways that I have encountered sine and cosine in my studies. There are no courses on trigonometry at my school, so perhaps that's why I feel like ...
7
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3answers
241 views

Laplace Equation with Tangential Derivative Prescribed on the Boundary

Consider the following Laplace boundary value problem (BVP) $$\matrix{ {{\nabla ^2}\Phi (x,y) = 0,} \hfill & { - a \le x \le a} \hfill & { - b \le y \le b} \hfill \cr {{{\partial \Phi ...
7
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1answer
225 views

Reduction to standard form.

I was wondering whether this ODE has been studied yet or whether there is anything we can say about its solutions? $$(1-t^2)u_{tt}-tu_t+4\left[n\beta (2t^2-1)+ \beta^2 (2t^2-1)^2+C\right]u=0$$ $C$ ...
7
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2answers
130 views

Integrating Factor - Not getting the answer given?

$y' - 4y = t$ My integrating factor is $e^{-4t}$ $\int e^{-4t}y'$ - $\int 4e^{-4t}y$ = $\int te^{-4t}$ $\int (e^{-4t}y)'$ = $\int te^{-4t}$ $e^{-4t}y$ = $-4te^{-4t}$ - $e^{-4t}$ I end up with ...
7
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3answers
928 views

Solving separable differential equation

Seems straight-forward but I've been unable to get it right. Here are my steps: $$y'(x) = \sqrt{-2y(x) + 28},\hspace{20 pt} y(-4)=-4$$ $$\int {1 \over \sqrt{28-2y} }\hspace{2 pt}\text{d}y = \int ...
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2answers
111 views

Intuition of Gronwall lemma

The Gronwall lemma is a well known and very useful statement which is used in many situations, in particular in the theory of differential equations. I have seen it so many times and even the proof is ...
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2answers
170 views

What is the general definition of time-scale for a differential equation?

For a differential equation like $\dot{x}=ax$ where $x$ is a function on an independent variable $t$, and $\dot{x}=\frac{dx}{dt}$, and $a$ is a constant, we define the time-scale $\frac{1}{a}$, which ...
7
votes
1answer
153 views

My first partial differential equation attempt

have I solved this correctly? My textbook is asking for the relation between $ \alpha $ and $ \beta $: $$ \frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}} $$ Textbook's proposed ...
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2answers
188 views

Can't seem to solve this differential equation

Disclaimer: This IS homework. So I will outline the steps I've taken an where I'm stuck. I have the following DE: $$ xy' = y + x\cos^2\left(\frac{y}{x}\right) $$ I then rule out the possible ...
7
votes
1answer
462 views

Is there a basis-independent proof of Abel's identity?

Abel's identity states that if $X(t)$ and $A(t)$ are $n\times n$ matrix-valued functions such that $X'(t)=A(t)X(t)$, then $\frac{d}{dt}(\det X(t)) = \mathrm{tr}\,A(t) \cdot \det X(t)$. The ...
7
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1answer
149 views

Using Poincaré-Bendixson to prove that there is a periodic solution

I want to use the Poincaré-Bendixson theorem to show that there exists a nontrivial (and periodic) solution to $$z'' + [\log (z^2 +4(z')^2)]z' + z = 0.$$ Therefore I substituted $u = z'$ to get $$u' ...
7
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1answer
194 views

Properties of the solutions to $x'=t-x^2$

Let $f_c$ be the solution to $$ \left\{ \begin{array}{c} x'=t-x^2 \\ x(0) =c \end{array} \right. $$ I'm trying to prove: If $c \geq 0$ then $f_c(t)$ is defined for all $t>0$ There is a ...
7
votes
1answer
542 views

How do you solve $f'(x) = f(f(x))$?

A friend told me to solve the following differential equation: $$f'(x)=f(f(x))$$ I have no idea how to solve this! This doesn't seem to be an ordinary differential equation and I can't even solve ...
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3answers
125 views

Integral formulation for the solution of $xy'' + y' = y$

Let's say that $y$ satisfies the following ODE: $$xy'' + y' = y$$ I want to formulate $y$ as a contour integral. I know that the final result I should get is: $$y(x)=\frac{1}{2i\pi} ...
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1answer
146 views

Reference request: a differential equation arising in geometry

$$ \frac{d\beta}{d\alpha} = \frac {\sin\beta}{\sin\alpha} $$ In what contexts (if any) is this equation known to occur?
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1answer
197 views

What's the name of this chaotic system? (Cool pics included.)

I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a ...
7
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1answer
293 views

$f(x)=\int_{0}^{+\infty} e^{-(t+\frac{1}{t})x}dt$ how to find $f(x)$?

$$f(x)=\int_0^{+\infty} e^{-(t+\frac{1}{t})x}\;dt$$ if while $ x>0 $ , $ f(x) $ has values I noticed some interesting relations for $f(x)$ as shown below: $$ \begin{align} t & ...
7
votes
1answer
259 views

A higher-order differential equation involving absolute values and trigonometry

For a smooth function $f: (-\pi/2,\pi/2) \to \mathbb{R} $, if $\displaystyle\frac{|f''(x)|}{\sqrt{(1+f'(x))^3}} = \cos{x}$, and $f(0) = f'(0) = 0$, $f''(0) = 1$, ...
7
votes
2answers
458 views

Prove that a map is continuous

Let $r:[0,1]\to\mathbb R$ be a continuous function and let $u_\lambda$ be the unique solution of the Cauchy Problem: $$\begin{cases}u''(t)+\lambda r(t)u(t)=0,\quad\forall t\in [0,1],\\ ...
7
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2answers
223 views

Why is it wrong to derive the chain rule this way?

My book says that the chain rule can stated as $$\dfrac{dy}{dx} = \dfrac{dy}{dt} \dfrac{dt}{dx}$$ However, it the book says that it is incorrect to reason that the chain rule is true because the ...