Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Solve an undamped oscillator and verify conserved energy.
- Justify the conclusion "x=cos 2t+sin 2t; E=(x′)²/2+2x²=4" using the stated assumptions.
Before you start
Second derivatives and trigonometric functions.
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
Solve x″+4x=0 with x(0)=1 and x′(0)=2.
Why this math matters
Solve an undamped oscillator and verify conserved energy. 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:
- There is no forcing or damping.
- Coefficients and units are normalized consistently.
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.
Use two initial conditions in a second-order motion model
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve x″+4x=0 with x(0)=1 and x′(0)=2.
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
x=A cos 2t+B sin 2t
These two independent solutions span the homogeneous solution space.
Work through the mathematics
A=1 and 2B=2, so B=1
Position and velocity supply separate initial conditions.
Check the conclusion
x=cos 2t+sin 2t; E=(x′)²/2+2x²=4
Differentiating energy gives x′(x″+4x)=0, so its initial value is conserved.
The result
x=cos 2t+sin 2t; E=(x′)²/2+2x²=4
Differentiating energy gives x′(x″+4x)=0, so its initial value is conserved.
Common mistakes to catch
- The coefficient four is not the angular frequency itself.
- A second-order ODE generally needs two initial conditions.
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 angular frequency?
Show a hint
Compare with x″+ω²x=0.
Reveal answer and explanation
Two
The coefficient four is ω².
Practice 2
What extra term models simple linear damping?
Show a hint
Use a force proportional to velocity.
Reveal answer and explanation
A positive multiple of x′ on the left
It makes the corresponding mechanical energy nonincreasing.
Take the idea with you
Separate oscillation frequency, phase, and amplitude in a vibration model.
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: Check whether a numerical decay method actually decays
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