A piecewise function has the following features:

2022-02-12

A piecewise function has the following features:

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asked 2021-09-09

Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations P(−3, −6, −1), Q(−1, −9, −8).

asked 2022-04-03

asked 2022-03-31

How to solve ${X}_{xx}=(1+\delta \left\{\left(x\right)\right\})X$

I am trying to find the vibration modes of a string that has a uniform mass density, plus some point mass somewhere attached to it, modelled by an additional Dirac delta function in the mass density. The wave equation is of the form

${u}_{xx}=(1+\delta \left(x\right)){u}_{tt}$

where u is the deformation, and $(1+\delta \left(x\right))$ the mass density. After separation of variables we find

${X}_{xx}=(1+\delta \left(x\right))X$

where X is the spatial part of the solution. Is there any analytical solution for X?

asked 2022-04-28

Limit for $\to \mathrm{\infty}$ of a solution to an ODE

Consider the Cauchy problem:

$\{\begin{array}{l}{y}^{\prime}(t)=\frac{{t}^{2}}{2+\mathrm{sin}({t}^{2})}cos(y(t{)}^{2})\\ y(0)=0\end{array}$

asked 2022-05-16

Find the indefinite integral and check the results by differentiation.

$\int \frac{dx}{{x}^{7}}$

$\int \frac{{x}^{4}+3{x}^{2}-5}{{x}^{3}}dx$

asked 2022-05-16

Evaluate the following:

${\int}_{2}^{5}(4{x}^{3}-8x+7)dx$

${\int}_{o}^{-\mathrm{\pi}}(5\mathrm{cos}\theta d\theta )$

asked 2022-03-27

Continuing Solutions of $\dot{x}=\frac{{t}^{2}{x}^{5}}{1+{x}^{2}+{x}^{4}}$ to entire number line

can be continued to the whole real line.

I know that this ODE is seperable as follows

$\frac{1+{x}^{2}+{x}^{4}}{{x}^{5}}dx={t}^{2}dt$

Thus, giving the solution

$-\frac{1}{4{x}^{4}}-\frac{1}{2{x}^{2}}+\mathrm{ln}\left(\left|x\right|\right)+C={\frac{13}{t}}^{3}$

However, forom here, it is not clear to me how any solution x(t) can be continued to the entire real number line.