Week 3 · From Linearity to the Laplace Transform

English translation

Before practice

01 Use known responses

If you know the motion caused by an applied force, can you predict the motion when its magnitude changes or several forces act together?

Description

Superposition applies to linear models
Match conditions when comparing input responses, for example by setting initial conditions to 0

02 Why express an equation in another form?

A differential equation describes changes over time. Could expressing the same relation in another form make calculations or system connections easier?

Description

Convert time relations into algebraic relations in the s-domain
Use for model connections and response calculations. Account for initial conditions separately.

Practice

1. Counterexample to linearity

Determine the linearity and time invariance of $3x$ , $x^2$ , $3x+1$ , $\cos(t)x$ , and $\frac{dx}{dt}$ , and explain your reasoning for each.

2. Linear Approximation

Keep the notebook's $x$ range and spacing, and calculate $|\sin x-x|$ at each $x$ . Select samples whose error is at most $0.01$ , then record the endpoint $x$ values of the contiguous interval containing $0$ .
For $x\in\{0.1,0.5,1\}$ rad, tabulate $\sin x$ , the approximation $x$ , absolute error $|\sin x-x|$ , and whether the tolerance is met.

3. Laplace Transform

$u(t)$ is a unit step that changes from 0 to 1 at $t=0$ and stays there.

Check the transform and inverse transform of $e^{-2t}$ and sin(4t), then solve $\ddot x+4\dot x+3x=6u(t)$ with all initial conditions at 0.
Verify the solution by substituting it into the original equation.
Without the toolbox, decompose $X(s)=\frac6{s(s+1)(s+3)}$ into $\frac As+\frac B{s+1}+\frac C{s+3}$ to find the coefficients, then write the solution and its derivative in exponential form yourself and verify them.

Submission and saving

Include code, actual outputs and plots, and your own interpretation. Submit through KLAS; check the semester schedule and latest announcements for the deadline.

Filename: Name_StudentID_Week3.mlx.

Restart the runtime, run all code from the top, then save.

Optional practice

Example: Passing a value in degrees to a radian input

Examine using sin(30) to calculate the sine of 30°. Record this result and sind(30).
If keeping sin, change the input line to sin(pi/6). Record the values before and after the change and the input unit.