Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Choose an integrating factor that combines two terms into one derivative.
- Justify the conclusion "y=2−exp(−2t)" using the stated assumptions.
Before you start
Product rule and exponential integration.
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 y′+2y=4 with y(0)=1.
Why this math matters
Choose an integrating factor that combines two terms into one derivative. 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:
- The initial time is zero.
- The coefficients are constant in the worked example.
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.
Solve a linear ODE by building a product derivative
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve y′+2y=4 with y(0)=1.
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
Multiply by exp(2t): exp(2t)y′+2exp(2t)y=4exp(2t)
The chosen factor makes the left side a product derivative.
Work through the mathematics
(exp(2t)y)′=4exp(2t) ⇒ exp(2t)y=2exp(2t)+C
Integrate both sides after recognizing the product.
Check the conclusion
y=2−exp(−2t)
The initial condition sets C=−1; substituting verifies both the ODE and the starting value.
The result
y=2−exp(−2t)
The initial condition sets C=−1; substituting verifies both the ODE and the starting value.
Common mistakes to catch
- Multiplying only one term by the integrating factor breaks the equation.
- The integration constant is fixed by the initial condition, not discarded.
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 long-time limit?
Show a hint
The transient exponential decays.
Reveal answer and explanation
Two
The solution approaches the equilibrium 2.
Practice 2
For y′+a(t)y=b(t), what factor is used?
Show a hint
Match its logarithmic derivative to a(t).
Reveal answer and explanation
exp(∫a(t)dt)
Its derivative supplies exactly the missing product-rule term.
Take the idea with you
Interpret a relaxing system as equilibrium plus a decaying transient.
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: Distinguish exact equilibrium from growth toward one
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