Solving DE with Laplace Transforms and g(t) Function

[s3a]

Homework Statement

Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms.

The g(t) function is a piecewise function and is attached as g.jpg.

Homework Equations

Laplace transforms of regular and unit step/Heaviside functions.

The Attempt at a Solution

My work is attached as MyWork.jpg. I suspect the part with the e to be the culprit but I'm not specifically sure as to what I did wrong and would appreciate it if someone could point it out to me.

Thanks in advance!

 

[LCKurtz]

Homework Statement

Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms.

The g(t) function is a piecewise function and is attached as g.jpg.

Homework Equations

Laplace transforms of regular and unit step/Heaviside functions.

The Attempt at a Solution

My work is attached as MyWork.jpg. I suspect the part with the e to be the culprit but I'm not specifically sure as to what I did wrong and would appreciate it if someone could point it out to me.

Thanks in advance!

I didn't check all your steps, but it looks to me you haven't taken into account the f(t) = t part. Your formula for that is$$
f(t) = t(1-u(8\pi))+8\pi u(t-8\pi)$$

 

[s3a]

What does u(8π) mean?

 

[Ray Vickson]

Homework Statement

Solve the differential equation y'' + 4y = g(t); y(0) = 2, y'(0) = 2 using Laplace transforms.

The g(t) function is a piecewise function and is attached as g.jpg.

Homework Equations

Laplace transforms of regular and unit step/Heaviside functions.

The Attempt at a Solution

My work is attached as MyWork.jpg. I suspect the part with the e to be the culprit but I'm not specifically sure as to what I did wrong and would appreciate it if someone could point it out to me.

Thanks in advance!

Why don't you just use the definition, rather than applying formulas that can be misused (as you did)? For $g(t) = \min(t,8 \pi)$ we have
$$L[g](s) = \int_0^\infty e^{-st} g(t) \, dt = \int_0^{8 \pi} e^{-st} t \, dt <br /> + \int_{8 \pi}^\infty e^{-st} 8 \pi \, dt,$$ and just do both integrals.

RGV

 

[LCKurtz]

What does u(8π) mean?

Sorry, that was a typo; that expression should be $u(t-8\pi)$ but the time for me to edit and correct it has expired.

 

[euquila]

u(t-8∏) is a unit step function. You can think of it as energy going into your system after 8∏ time has elapsed. The function is 0 <= 8∏ and 1 after 8∏.