opatovaL
2021-02-21
Answered

Find the Laplace transforms of the given functions.

$g\left(t\right)=4\mathrm{cos}\left(4t\right)-9\mathrm{sin}\left(4t\right)+2\mathrm{cos}\left(10t\right)$

You can still ask an expert for help

Dora

Answered 2021-02-22
Author has **98** answers

Step 1 : Introduction

Given function is,

$g\left(t\right)=4\mathrm{cos}\left(4t\right)-9\mathrm{sin}\left(4t\right)+2\mathrm{cos}\left(10t\right)$

We have to evaluate the Laplace transform of given function g(t)

step 2

$g\left(t\right)=4\mathrm{cos}\left(4t\right)-9\mathrm{sin}\left(4t\right)+2\mathrm{cos}\left(10t\right)$

using Laplace transform properties

$L\{\mathrm{sin}at\}=\frac{a}{{s}^{2}+{a}^{2}}$

$L\{\mathrm{cos}at\}=\frac{s}{{s}^{2}+{a}^{2}}$

Applying Laplace transform on g(t) function

$\therefore L\left\{g\left(t\right)\right\}=L\{4\mathrm{cos}\left(4t\right)-9\mathrm{sin}\left(4t\right)+2\mathrm{cos}\left(10t\right)\}$

$L\left\{g\left(t\right)\right\}=4L\{\mathrm{cos}\left(4t\right)\}-9L\{\mathrm{sin}\left(4t\right)\}+2L\{\mathrm{cos}\left(10t\right)\}$

$=4\left(\frac{s}{{s}^{2}+{4}^{2}}\right)-9\left(\frac{4}{{s}^{2}+{4}^{2}}\right)+2\left(\frac{s}{{s}^{2}+{10}^{2}}\right)$

$=\frac{4s}{{s}^{2}+16}-\frac{36}{{s}^{2}+16}+\frac{2s}{{s}^{2}+100}$

Thus , Laplace transform of given function g(t) is

$L\left\{g\left(t\right)\right\}=\frac{4s}{{s}^{2}+16}-\frac{36}{{s}^{2}+16}+\frac{2s}{{s}^{2}+100}$

Given function is,

We have to evaluate the Laplace transform of given function g(t)

step 2

using Laplace transform properties

Applying Laplace transform on g(t) function

Thus , Laplace transform of given function g(t) is

asked 2021-12-28

asked 2022-03-17

Calculate the laplace transform of

${t}^{2}u(t-2)$

I don't know how to manipulate ${t}^{2}$ in order for it to meet the form of the product between a function and a heaviside function.

asked 2022-01-20

How to solve $\frac{dy}{dx}=5xy+\mathrm{sin}x$ ?

With$y\left(0\right)=1$ . Use an integrating factor.

With

asked 2022-08-12

Why does the method of separable equations work in differential equations?

In differential equations, the idea of multiplying by an infinitesimal, dx, is used in the method of separable equations. My confusion is in why this works as I've heard many times in the past that you shouldn't multiply and cancel out infinitesimals like that. I understand that in the setting of nonstandard analysis that there may not be something wrong with this, but in the usual setting is there a more rigorous interpretation of this?

In differential equations, the idea of multiplying by an infinitesimal, dx, is used in the method of separable equations. My confusion is in why this works as I've heard many times in the past that you shouldn't multiply and cancel out infinitesimals like that. I understand that in the setting of nonstandard analysis that there may not be something wrong with this, but in the usual setting is there a more rigorous interpretation of this?

asked 2022-03-22

In this problem, $y={c}_{1}{e}^{x}+{c}_{2}{e}^{-x}$ is a two-parameter family of solutions of the second-order DE $y{}^{\u2033}-y=0$ . Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

$y\left(0\right)=1,\text{}{y}^{\prime}\left(0\right)=8$

asked 2022-01-19

Is the following is linear equation?

${\left(\frac{dy}{dx}\right)}^{2}+\mathrm{cos}\left(x\right)y=5$

asked 2021-05-23

Find the differential of each function.

(a)

(b)