Solve: $\frac{dy}{dx}-\frac{dx}{dy}=\frac{y}{x}-\frac{x}{y}$

hadejada7x
2022-01-20
Answered

Solve: $\frac{dy}{dx}-\frac{dx}{dy}=\frac{y}{x}-\frac{x}{y}$

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Laplace transform of $f\left(t\right)=t{e}^{-t}\mathrm{sin}\left(2t\right)$

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Solve the initial value problem

${x}^{(4)}-5x"+4x=1-{u}_{x}(t),$

$x(0)={x}^{\prime}(0)=x"(0)={x}^{\u2034}(0)=0$

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Applications of Differential and Difference Equations

Solve$({D}^{2}-2D+1)y=x{e}^{x}\mathrm{sin}x$

Solve

asked 2022-09-11

I want to find the Laplace transform of

$$f(t)={\int}_{0}^{t}\mathrm{sin}(r)u(r-\beta )(t-r{)}^{\alpha}dr$$

for $\alpha ,\phantom{\rule{thinmathspace}{0ex}}\beta >0$ where u is the unit step function. I don't understand what my approach should be here - I know that

$$\mathcal{L}\{u(t-c)f(t-c)\}={e}^{-sc}\mathcal{L}\{f(t)\},$$

but I don't see how to convert the expression with the integral into the familiar form

$$f(t)={\int}_{0}^{t}\mathrm{sin}(r)u(r-\beta )(t-r{)}^{\alpha}dr$$

for $\alpha ,\phantom{\rule{thinmathspace}{0ex}}\beta >0$ where u is the unit step function. I don't understand what my approach should be here - I know that

$$\mathcal{L}\{u(t-c)f(t-c)\}={e}^{-sc}\mathcal{L}\{f(t)\},$$

but I don't see how to convert the expression with the integral into the familiar form

asked 2022-09-24

If $f(x)=\sum _{n=0}^{\mathrm{\infty}}{a}_{n}{x}^{n}$ converges for all $x\ge 0$, show that $\mathcal{L}\{f\}(s)=\sum _{n=0}^{\mathrm{\infty}}\frac{{a}_{n}n!}{{s}^{n+1}}$

asked 2021-09-26

find Laplace transform

$L}^{-1}\frac{5}{{s}^{2}+2s-3$

asked 2021-03-04

Find the Laplace transformation (evaluating the improper integral that defines this transformation) of the real valued function f(t) of the real variable t>0. (Assume the parameter
s appearing in the Laplace transformation, as a real variable).

$f\left(t\right)=2{t}^{2}-4\mathrm{cosh}\left(3t\right)+{e}^{{t}^{2}}$