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

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Understanding proof of Peano's existence theorem

I'm studying the proof of Peano's existence theorem on this paper. At page 5 it is said that the problem $$\begin{cases} y(t) = y_0 & \forall t ∈ [t_0, t_0 + c/k] \\ y'(t) = f(t − c/k, y(t − ...
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0answers
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Solving Special Function Equations Using Lie Symmetries

The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's ...
3
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1answer
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Describing non-vanishing $1$-forms on two dimensional manifolds.

Let $h_1 \mathrm{d}x_1 + h_2 \mathrm{d}x_2$ be a non-vanishing $1$-form on a $2$-dimensional manifold. Why do locally exist smooth functions $f,g$ with $f\mathrm{d}g= h_1 \mathrm{d}x_1 + h_2 ...
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1answer
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Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= (\frac{\partial}{\partial t}\hat{\Phi}_t) \hat{\Phi}_t^{-1}) \\ &= (\frac{\partial}{\partial t}\hat{\Phi}_t) ...
1
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1answer
51 views
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A problem on infinite domain diffusion equation

Consider the following problem $$u_t-u_{xx}=p(x,t), -\infty<x<\infty,t>0$$ $$u(x,0)=0$$ $$u\rightarrow0 \text{ as } x\rightarrow \pm \infty$$ This can be solved using many sub problems as ...