# Tagged Questions

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### Find solutions for an differential equation system

I have a differential equation system $x_1'(t) = -x_2(t)$ $x_2'(t) = -x_1(t)$ I see that I can write $\dot{x} = Ax$ where $A = \begin{pmatrix}0 & -1 \\ -1 & 0\end{pmatrix}$ The complete ...
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### Simplifying a coupled-pendulum equation by assumption

I have been given the following question and I am unsure if I am missing an assumption or if I am misunderstanding something else: Two identitical pendula each of length $\ell$ and with bobs of ...
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### Arc length paramatrizations satisfy original system of differential equations?

Say we have a system of differential equations $$\begin{cases} x'''(t)+f(t)x'(t)=0\\ y'''(t)+f(t)y'(t)=0 \end{cases}$$ on an interval $[a,b]$, along with the restriction that $$x'(t)^2+y'(t)^2=1$$ ...
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### How do I solve this question without solving for the functions?

The problem goes as follows: \begin{aligned} \frac{d y_1}{dt} &= -ay_1 \\ \frac{d y_2}{dt} &= -by_2 -\frac{dy_1}{dt} \\ y_1(0)&=M \\y_2(0)&=0 \end{aligned} where ...
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### System of integral equations for a unimodal symmetric probability distribution

Let $f(x)$ be a symmetric unimodal probability distribution on $\mathbb R$, with mean $\mu=0$. By unimodal, I mean that $f(x)$ is strictly increasing for $x<\mu$ and strictly decreasing for ...
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