Tagged Questions

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

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Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$.

Given the function $$f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} & x \neq 0 \\ 0 & x = 0 \end{array}\right.$$ show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$. So I have to show ...
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Fourier Series for $|\cos(x)|$

I'm having trouble figuring out the Fourier series of $|\cos(x)|$ from $-\pi$ to $\pi$. I understand its an even function, so all the $b_n$s are $0$ $$a_0 = \frac 2 \pi \int_0^\pi |\cos(x)|\,dx = 0$$...
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Does this ODE question have closed form solution?

These days, I am struggling with following ODE problem when I build up my research model: $1/2f''(x)+a(b - x) f'(x) -(c+ e^{A+Bx})f(x)=0$ where f(x) is a smooth function, and $a,b,c, A,B$ ...
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How to Solve the Coupled Differential Equations?

I came across the set of following coupled equations while studying cycloid motion in Griffiths' Intro to ED $\ddot{y}=\omega \dot{z}$ $\ddot{z}=\omega (\frac{E}{B}-\dot{y})$ I am at a loss as to ...
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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 ...
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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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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 ...
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Function whose third derivative is itself.

I'm looking for a function $f$, whose third derivative is $f$ itself, while the first derivative isn't. Is there any such function? Which one(s)? If not, how can we prove that there is none? Notes: ...
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Can this gravitational field differential equation be solved, or does it not show what I intended?

This is the equation I'm having trouble with: $$G \frac{M m}{r^2} = m \frac{d^2 r}{dt^2}$$ That's the non-vector form of the universal law of gravitation on the left and Newton's second law of ...
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Fourth Order Nonlinear ODE

I was looking at an ode $w^{(4)} + w^3 = 0$ with initial conditions $[w'''(0),w''(0),w'(0),w(0)]=[1,0,0,0]$. I can see via maple that there is a blowup around 3.7. I was wondering if there was a way ...
For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ... 4answers 11k views Can someone intuitively explain what the convolution integral is? I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ... 2answers 2k views Particular solution to a Riccati equation y' = 1 + 2y + xy^2 The equation is y' = 1 + 2y + xy^2. I've tried mx+n, ax^m, even \tan x as candidates for particular solution where a,m,n \in \mathbb Q, but it did not work. Can anyone find one particular ... 3answers 18k views Definition of a Differential Equation? Here is one definition of a differential equation: "An equation containing the derivatives of one or more dependent variables, with respect to one of more independent variables, is said to be a ... 1answer 297 views Finite dimensional spaces What are the finite-dimensional spaces W of differentiable functions with this property: If f is in W, then \frac{df}{dx} is in W. 1answer 880 views Recursive solutions to linear ODE. When finding the solutions to the simple ODE$$ y'- mxy= x^n \text{ ; } y(0) = 0$$I found the following: Let P_n be the particular solution for each integer exponent n. Then if we define$$...
Let $f$ be a real polynomial of two variables. Let $\partial_f=f\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)$. Let $H$ denote the space of harmonic polynomials, i.e., ...