# Forced response and Laplace transform

## Homework Statement

d'^2 (y)/dt + 4 (dy/dt) + 4y = -7(e^(-3t)). Here I need to forced response of this differential equation using laplace transform technique.

## The Attempt at a Solution

I understand the part of converting each term to each laplace,
d^2y/dt to Y(s)*s^2, dy/dt to Y(s)*s, y(t) to Y(s), where each term is being converted from f(t) to F(s). I really confused on how to proceed with this question form here LCKurtz
Homework Helper
Gold Member

## Homework Statement

d'^2 (y)/dt + 4 (dy/dt) + 4y = -7(e^(-3t)). Here I need to forced response of this differential equation using laplace transform technique.

## The Attempt at a Solution

I understand the part of converting each term to each laplace,
d^2y/dt to Y(s)*s^2, dy/dt to Y(s)*s, y(t) to Y(s), where each term is being converted from f(t) to F(s). I really confused on how to proceed with this question form here Unless you have a couple of 0 initial conditions, you need to put them in the transforms of the derivatives. You also need to transform the right side of the DE. Solve the resulting equation for Y(s) and invert it.

I found out by applying laplace to each factor,

s^2y(t)+4sy(t)+4y(t)=-7t(t+3)
s^2y(t)+4sy(t)+4y(t)=-7t(t)-7y(3)
y(t)(s^2+4s+11)=y(3)

But from here, I am confused. Since in the question the value of u(t) was given to be e^-3t. Other wise I would have used laplace transform of u(t) which is 1/s.
Also in the last line, I couldn't factorize the polynomial which is hindering my progress.
Any help would be whole heartedly appreciated.

LCKurtz
Homework Helper
Gold Member

## Homework Statement

d'^2 (y)/dt + 4 (dy/dt) + 4y = -7(e^(-3t)).

Unless you have a couple of 0 initial conditions, you need to put them in the transforms of the derivatives. You also need to transform the right side of the DE. Solve the resulting equation for Y(s) and invert it.

I found out by applying laplace to each factor,

s^2y(t)+4sy(t)+4y(t)=-7t(t+3)

Did you read my reply? Are your initial conditions y(0)=0 and y'(0) = 0 or not? You were using Y(s) for the transform of y(t). You have t's in your transform when they should be s's and y when you should have Y. And where did the t(t+3) on the right side come from? It surely isn't the transform of e^(-3t).

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

d'^2 (y)/dt + 4 (dy/dt) + 4y = -7(e^(-3t)). Here I need to forced response of this differential equation using laplace transform technique.

## The Attempt at a Solution

I understand the part of converting each term to each laplace,
d^2y/dt to Y(s)*s^2, dy/dt to Y(s)*s, y(t) to Y(s), where each term is being converted from f(t) to F(s). I really confused on how to proceed with this question form here L[y'(t)](s) ≠ s*Y(s) and L[y"(t)](s) ≠ s^2*Y(s) in general (although for some special conditions on y(.) these are true). Check your sources!

RGV