- #1

Aristotle

- 169

- 1

## Homework Statement

Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..)

y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0

## Homework Equations

Lf'' = ((s^2)*F) - s*f(0) - f'(0)

Lf' = sF - f(0)

Lf = F(s)

## The Attempt at a Solution

My first attempt is of course realizing that the above equation is expressed in terms of 't'. So I must take the laplace transform on both sides.L(y'') + 2 L(y') + 2L(y) = L(2)

[s^2*Y(s) - sy(0) - y'(0) ] + 2 [ sY(s) - y(0) ] + 2*Y(s) = 2/s

[s^2*Y(s) - 0 - 0 ] + 2 [ sY(s) - 0 ] + 2*Y(s) = 2/s

s^2*Y(s) + 2s*Y(s) + 2*Y(s) = 2/s

Y(s) [s^2 + 2s + 2] = 2/s

Y(s) = 2/s * (1/ [s^2 + 2s + 2] )

= 2/s * (1 / [(s+1)^2 + 1]) (using the formula (s+(a/2)^2) + b - (a^2)/4 )

At this final step, I am lost on how to break the equation apart to solve for Y(t) using inverse laplace transform without using "Partial Fractions".

Can somebody guide me to the right direction? Thank you.

The solution according to my professor is:

y = -e^-t * sin(t) - e^-t*cos(t) + 1

Last edited: