Solving differential equation with Laplace transforms using partial fractions

Cocoleia · Jan 18, 2018

Homework Statement

I am given this equation:

$Differential equation d²x/dt² + 6 dx/dt + 8x = 5 sin 3t, with the derivative terms written in fraction notation.$

and asked to solve using Laplace transforms

The Attempt at a Solution

This is what I did:

$Handwritten derivation solving s^2 X(s) + 6s X(s) + 8 X(s) = 15/(s^2+9) for X(s), factoring to X(s) = 15/((s-2)(s-4)(s-3)(s+3)) and a partial fraction setup 15/(...) = A/(s-2) + B/(s-4) + C/(s-3) + D/(s+3).$

$Handwritten partial fraction decomposition on lined paper: X(s) = 1/(s-2)... solved by cover-up values for A, B, C, D, then inverse Laplace transform e^(2t)u(t), e^(4t)u(t), e^(3t)u(t) terms.$

This seemed logical to me, I used partial fractions and it stayed pretty simple.

This is what the solutions my prof posted do:

$Handwritten-style worked solution of a Laplace transform problem: (s^2+6s+8)X(s) = 15/(s^2+9), solved for X(s) = 15/((s^2+9)(s^2+6s+8)), partial fraction expansion, and the inverse transform x(t) = -18/65 cos(3t) - 1/65 sin(3t) - 3/10 e^(-4t) + 15/26 e^(-2t).$

Is my answer equivalent to this, or are their faults in my logic and work ?

gneill · Jan 18, 2018

One problem I can see with your work is where you write:

$Handwritten equation: 15/(s^2 + 9) = 15/((s - 3)(s + 3)), with an equals sign between the two fractions.$

##(s^2 + 9) \neq (s - 3)(s + 3)##

rude man · Jan 19, 2018

Also,
s^2 + 6s + 8 does not factor to (s-2)(s-4). Poles in the rt-hand plane are bad business!

rude man · Jan 19, 2018

Cocoleia said:

Homework Statement

I am given this equation:
View attachment 218670
and asked to solve using Laplace transforms

The Attempt at a Solution

This is what I did:
View attachment 218671
View attachment 218672
This seemed logical to me, I used partial fractions and it stayed pretty simple.

This is what the solutions my prof posted do:
View attachment 218673

Is my answer equivalent to this, or are their faults in my logic and work ?

Their (sic) are faults in your work ...

Solving differential equation with Laplace transforms using partial fractions

Homework Statement

The Attempt at a Solution

Discussion

Homework Statement

The Attempt at a Solution

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