Lewis Harvey
2021-01-22
Answered

Use Laplace transform to solve the folowing initial value problem $y"+2{y}^{\prime}+y=4{e}^{-t}y(0)=2{y}^{\prime}(0)=-1$

You can still ask an expert for help

yunitsiL

Answered 2021-01-23
Author has **108** answers

Step 1

Given :$y"+2{y}^{\prime}+y=4{e}^{-t}y(0)=2{y}^{\prime}(0)=-1$

Applying Laplace Transform

$L(y")+2L({y}^{\prime})+L(y)=4L({e}^{(}t))$

$\Rightarrow {s}^{2}Y(s)-sy(0)-{y}^{\prime}(0)+2[sY(s)-y(0)]+Y(s)=\frac{4}{(s+1)}$

$\Rightarrow ({s}^{2}+2s+1)Y(s)-2s+1-4=\frac{4}{(s+1)}[y(0)=2,{y}^{\prime}(0)=-1]$

$\Rightarrow (s+1{)}^{2}Y(s)=\frac{4}{(s+1)}+2s+3$

$\Rightarrow Y(s)=\frac{4}{(s+1{)}^{3}}+\frac{(2s+3)}{(s+1{)}^{2}}$

$\Rightarrow Y(s)=\frac{4}{(s+1{)}^{3}}+\frac{2(s+\frac{3}{2}+1-1)}{(s+1{)}^{2}}$

$\Rightarrow Y(s)=\frac{4}{(s+1{)}^{3}}+\frac{2(s+1)}{(s+1{)}^{2}}+\frac{1}{(s+1{)}^{2}}$

$Y(s)=\frac{4}{(s+1{)}^{3}}+\frac{2(s+1)}{(s+1{)}^{2}}+\frac{1}{(s+1{)}^{2}}$

Step 2

Now taking inverse Laplace Transform,

${L}^{-1}\{Y(s)\}=4{L}^{-1}\left\{\frac{4}{(s+1{)}^{3}}\right\}+2{L}^{-1}\left\{\frac{2(s+1)}{(s+1{)}^{2}}\right\}+{L}^{-1}\left\{\frac{1}{(s+1{)}^{2}}\right\}$

Using the identity${L}^{-1}\left\{\frac{1}{(s+a{)}^{n}}\right\}=\frac{{t}^{n-1}{e}^{-at}}{(n-1)!}$ , we get

$y(t)=2{t}^{2}{e}^{-t}+2{e}^{-t}+t{e}^{-t}$

Given :

Applying Laplace Transform

Step 2

Now taking inverse Laplace Transform,

Using the identity

asked 2020-12-07

Solve the linear equations by considering y as a function of x, that is,

asked 2021-02-12

On solution of the differetial equation ${y}^{\u2033}+{y}^{\prime}=0$ is $y={e}^{-x}$ . Use Reduction of Order to find a second linearly independent solution.

asked 2022-01-18

Finding eigenvalues by inspection?

I need to solve the following problem,

In this problem, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of the system.

$x}_{1}^{{}^{\prime}}=2{x}_{1}+{x}_{2}-{x}_{3$

$x}_{2}^{{}^{\prime}}=-4{x}_{1}-3{x}_{2}-{x}_{3$

$x}_{3}^{{}^{\prime}}=4{x}_{1}+4{x}_{2}+2{x}_{3$

Now I know how to find the eigenvalues by using the fact that$|A-\lambda I|=0$ , but how would I do it by inspection? Inspection is easy for matrices that have the sum of their rows adding up to the same value, but this coefficient matrix doesn't have that property.

EDIT: Originally I didn't understand what inspection meant either. After googling it this is what I found. Imagine you have the matrix,$$A=(\begin{array}{ccc}2& -1& -1\\ -1& 2& -1\\ -1& -1& 2\end{array})$$

By noticing (or inspecting) that each row sums up to the same value, which is 0, we can easily see that [1, 1, 1] is an eigenvector with the associated eigenvalue of 0.

I need to solve the following problem,

In this problem, the eigenvalues of the coefficient matrix can be found by inspection and factoring. Apply the eigenvalue method to find a general solution of the system.

Now I know how to find the eigenvalues by using the fact that

EDIT: Originally I didn't understand what inspection meant either. After googling it this is what I found. Imagine you have the matrix,

By noticing (or inspecting) that each row sums up to the same value, which is 0, we can easily see that [1, 1, 1] is an eigenvector with the associated eigenvalue of 0.

asked 2020-10-18

Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by

where we assume s is a positive real number. For example, to find the Laplace transform of

Verify the following Laplace transforms, where u is a real number.

asked 2020-12-30

Find the inverse of Laplace transform

$\frac{2s+3}{(s-7{)}^{4}}$

asked 2020-11-20

Solve the linear equations by considering y as a function of x, that is,

asked 2022-01-20

Question about solving a differential equation

$D(D-3)(D+4)\left[y\right]=0$ ,

where D is the differential operator, how to get the general solution of y? The solution suggest that it is

$y=6{c}_{1}-2{c}_{2}\mathrm{exp}(-4t)+{3}_{{c}_{3}}\mathrm{exp}\left(3t\right)$

where D is the differential operator, how to get the general solution of y? The solution suggest that it is