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

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Convolution: How to construct it for a given function?

While working on my thesis my advisor handed me an unfinished paper which states the following: First, define the operators \begin{align*} A_i &:= -\operatorname{div}(\sigma_i\nabla) \\ A_e ...
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Initial Value Problem for $(\cos x -x\sin x +y^2)dx + 2xy\,dy =0$, $y(\pi )=1$

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following question. If it's any indication of difficulty, the exercise is only Part 1-B of the sheet, ...
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1answer
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Solving or knowing something about a non-linear PDE which is “almost” linear?

Let $a>0$ be fixed. I have the following PDE: $u=u(t,x)$, $t\in [0,1]$, $x\in \mathbb{R}$, $$-\partial_t u = |\partial_x u| + \frac{1}{2}\partial_x^2 u, \quad ...