Search Results

Think of it in terms of causality & superposition. $$z = f(x,y)$$ If you keep $y$ fixed then $\frac{dz}{dt} = \frac{df}{dx} * \frac{dx}{dt}$ If you keep $x$ fixed then $\frac{dz}{dt} = \frac{df}{fy …

A state-variable kind of approach: (represents the physical states of a system) Newton's first law of motion says that without external forces, masses will move at constant velocity. Every change in …

$$\begin{align*} \sum F &= ma\\ \frac{dv}{dt} &= a\\ &= \frac{\sum F}{m}\\ \sum F &= mg - kv\\ \frac{dv}{dt} &= g - \frac{k}{m} v\end{align*}$$ This is a differential equation with a solution of $$\ …

Try division? $\dfrac {1}{1-x} = 1 + x + x^2 + x^3 ...$ (edit: I'm not sure why you would try to derive it from the series for $(1+x)^n$ )

Partial fraction expansion gives $$ \frac{1}{u^{4} + \left(4 \zeta^{2} - 2\right)u^{2} + 1} = \frac{1}{4\zeta\sqrt{\zeta^2-1}}\left(\frac{1}{u^2+a_1{}^2} - \frac{1}{u^2+a_2{}^2}\right) $$ and since $ …

I am trying to compute $$I(\zeta) = \int_{-\infty}^{\infty} \frac{1}{u^{4} + \left(4 \zeta^{2} - 2\right)u^{2} + 1}\, du$$ for positive real $\zeta$. Can anyone help? I'm way out of practice for inte …

$\displaystyle\frac{\sin(x)}{x}$ is useful; it has a singularity at $x=0$, but if you take the union of $\displaystyle y=\frac{\sin(x)}{x}$ with the point $(x=0,y=1)$ then you get $\text{sinc}(x/\pi)$ …