# Tagged Questions

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

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### Differential Equation of the Form $\frac{dy}{dx}=\sin(x+y)$ [duplicate]

I have been attempting to solve the above differential equation for some time now, and I remain stuck on one step. After substituting $u=x+y$, separating the variables, and integrating both sides, I ...
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### Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
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### Eigenfunctions of the Laplacian

I am willing to offer a bounty for this one, so I will give you an exact idea of what I need: I am looking for solutions of $$\Delta \Psi(r,\theta)=k^2\Psi(r,\theta)$$ where $k\in \mathbb{R}$. Such ...
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### $x''= \frac{Ax+B}{Cx+D}$

Might there be a closed-form solution to the second-order differential equation below?$$x''(t)=\frac{Ax+B}{Cx+D}$$ If not, is there any way to get a power series approximation in terms of the ...
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### What comes after Differential Equations?

First of all, please do excuse the lack of correct terminology, I've haven't learnt Differential Equations at school (yet) so this question comes from just a bit of research I did for my own enjoyment....
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### A Differential Equation with Trigonometric Coefficients

Suppose we have the following second-order differential equation: $\cos^2(x)y'' -\sin(x)y' + y = 0$ How do we determine its general solution? I couldn't even guess a particular solution; all my ...
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### Restricted Three-Body Problem

The movement of a spacecraft between Earth and the Moon is an example of the infamous Three Body Problem. It is said that a general analytical solution for TBP is not known because of the complexity ...
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### How to estimate solutions to an ODE with an asymptotically nilpotent coefficient?

Suppose $f:\mathbb R\to\mathbb R^n$ satisfies $$f'(t) = A(t)f(t),$$ where $A$ is a smooth matrix-valued function. If I know that the matrix $A(t)$ is asymptotically nilpotent, how could I prove a ...
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### Stochastic Calculus Question

I'm new here and was hoping someone could help me answer this question. I'm reading a paper and I'm a bit confused on how they go from 1 equation to the next. They say: Let \begin{align} x(t) = {} &...
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### First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no friends/...
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### What condition ensures the solution is periodic? (ODE)

Suppose that $\phi$ is a solution to the ODE \begin{align} x' = f(x) + \sin(t) \end{align} What condition can we put on $f$ to ensure that $\phi$ is periodic of period $2\pi$? That is, what do we ...
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### An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
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### solving the integral $\int_0^\infty \dfrac{\sin xt}{x(x^2+1)} dx$

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