Use the definition of Laplace Transforms to find

Then rewrite f(t) as a sum of step functions,

Braxton Pugh
2021-02-12
Answered

Use the definition of Laplace Transforms to find

Then rewrite f(t) as a sum of step functions,

You can still ask an expert for help

Sadie Eaton

Answered 2021-02-13
Author has **104** answers

Step 1

Consider the given function:

Step 2

Now, by definition of Laplace Transform:

Step 3

Now, Rewrite f(t) as the sum of step functions,

Step 4 Now, use the following formula for Laplace transform of step function:

Step 5

Thus, now take Laplace transform:

Step 6

Thus, from equation (1) and (2), it can be seen that the Laplace transform of f(t) that is obtained from both the methods is same that is :

asked 2022-06-29

The theorem of existence and uniqueness is: Let ${y}^{\prime}+p(x)y=g(x)$ be a first order linear differential equation such that p(x) and g(x) are both continuous for $a<x<b$. Then there is a unique solution that satisfies it.

When a differential equation has no solution that satisfies $y({x}_{0})={y}_{0}$, what does this mean?? Can the theorem be verified??

When a differential equation has no solution that satisfies $y({x}_{0})={y}_{0}$, what does this mean?? Can the theorem be verified??

asked 2021-09-24

Use Laplace transform to solve the following initial value problem:

A)

B)

C)

D)

E) non of the above

F)

asked 2022-06-11

I would like some help on comprehending this question as well as a push in the right direction. The question gave a system of first-order differential equation.

$x(t{)}^{\prime}=4x(t)-3y(t)+6{e}^{2t}$

$y(t{)}^{\prime}=4x(t)-6y(t)$

The question asked me to find the 2nd inhomogeneous equation that satisfies x(t). Does this mean the answer should all be in terms of x? I tried focusing on the x and differentiating it with respect to t.

so $x(t{)}^{\prime}=4x(t)-3y(t{)}^{\prime}+6{e}^{2t}$ becomes:

$x(t{)}^{\u2033}=4x(t{)}^{\prime}-3y(t{)}^{\prime}+12{e}^{2t}$

for simplicity sake, I will write x(t) as x and y(t) as y.

After that step, I replaced the y' in the x'' equation with the rearranged y' from the original question into the differentiated x' equation. This gives:

${x}^{\u2033}=4{x}^{\prime}-3(4x-6(\frac{1}{-3}({x}^{\prime}-4x-6{e}^{2t})+12{e}^{2t}$

this cancels down to:

${x}^{\u2033}=4{x}^{\prime}-12x+2{x}^{\prime}-8x$

but if you move everything to one side, it becomes

${x}^{\u2033}-6{x}^{\prime}+20x=0$

this is a second-order homogenous equation, so I don't quite know where I went wrong

$x(t{)}^{\prime}=4x(t)-3y(t)+6{e}^{2t}$

$y(t{)}^{\prime}=4x(t)-6y(t)$

The question asked me to find the 2nd inhomogeneous equation that satisfies x(t). Does this mean the answer should all be in terms of x? I tried focusing on the x and differentiating it with respect to t.

so $x(t{)}^{\prime}=4x(t)-3y(t{)}^{\prime}+6{e}^{2t}$ becomes:

$x(t{)}^{\u2033}=4x(t{)}^{\prime}-3y(t{)}^{\prime}+12{e}^{2t}$

for simplicity sake, I will write x(t) as x and y(t) as y.

After that step, I replaced the y' in the x'' equation with the rearranged y' from the original question into the differentiated x' equation. This gives:

${x}^{\u2033}=4{x}^{\prime}-3(4x-6(\frac{1}{-3}({x}^{\prime}-4x-6{e}^{2t})+12{e}^{2t}$

this cancels down to:

${x}^{\u2033}=4{x}^{\prime}-12x+2{x}^{\prime}-8x$

but if you move everything to one side, it becomes

${x}^{\u2033}-6{x}^{\prime}+20x=0$

this is a second-order homogenous equation, so I don't quite know where I went wrong

asked 2021-02-13

Use a Laplace transform to determine the solution of the following systems with differential equations

a)${x}^{\prime}+4x+3y=0\text{with}x(0)=0$

${y}^{\prime}+3x+4y=2{e}^{t},y(0)=0$

a)

asked 2021-01-08

Solve differential equation${y}^{\prime}-2y=1-2x$

asked 2021-12-31

asked 2022-04-13

Laplace transform involving step function

$f\left(t\right)=\frac{\mathrm{sin}\left(2t\right)}{{e}^{2t}}+t\xb7u(t-4)$