Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Solve a linear steady sinusoidal response using phasors.
- Justify the conclusion "y=(2cos 3t+3sin 3t)/13" using the stated assumptions.
Before you start
Complex arithmetic and first-order differential equations.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For y′+2y=cos(3t), find the steady periodic solution.
Why this math matters
Solve a linear steady sinusoidal response using phasors. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Only the steady periodic response is requested.
- The equation has constant real coefficients.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Convert a forced oscillation into a complex amplitude
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For y′+2y=cos(3t), find the steady periodic solution.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Write the forcing as Re(e^(3it)) and try y=Re(Ce^(3it))
A complex exponential packages amplitude and phase into one constant.
Work through the mathematics
(2+3i)C=1 ⇒ C=(2−3i)/13
Solving the algebraic equation accounts for the derivative multiplier 3i.
Check the conclusion
y=(2cos 3t+3sin 3t)/13
Taking the real part gives the periodic response; a separate Ce^(−2t) transient may be added for an initial condition.
The result
y=(2cos 3t+3sin 3t)/13
Taking the real part gives the periodic response; a separate Ce^(−2t) transient may be added for an initial condition.
Common mistakes to catch
- Do not discard the transient when an initial condition matters.
- Complex amplitude is not itself the physical real-valued response.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the response amplitude?
Show a hint
Take the modulus of 1/(2+3i).
Reveal answer and explanation
1/√13
The denominator has modulus √(4+9).
Practice 2
Does the periodic solution determine every initial-value solution?
Show a hint
Consider the homogeneous equation.
Reveal answer and explanation
No
An exponentially decaying homogeneous term adjusts the initial condition.
Take the idea with you
Interpret phase lag and amplitude reduction in a simple first-order filter.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Generate polynomial identities from cosine addition
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