1. Integrate first: Find the general formula ($+ C$).
2. Identify the Witness: Write down the $(x, y)$ point given in the problem.
3. Solve for C: Plug $x$ and $y$ into your formula and find the secret constant.
4. The Particular Path: Write your final answer with the real number for $C$.
Find the **Particular Solution** for each differential equation.
$y' = 4x$. The curve passes through $(1, 10)$.
$y' = \cos x$. The curve passes through $(0, 5)$.
$y' = e^x$. The curve passes through $(0, 100)$.
The Falling Arrow: The velocity of an arrow is $v(t) = -32t + 100$.
Its initial height (at $t=0$) is 6 feet.
Find the height function $h(t) = \int v(t) dt$.
The Rebuilt Wall: The rate of building a wall is $R(x) = 3x^2 + 10$.
After 2 hours ($x=2$), the wall was 20 feet high.
How high was the wall when they started ($x=0$)?
If you have two different people with the same speed ($y' = 10$), but one has $C=5$ and the other has $C=50$... will they ever meet? Will the distance between them ever change? Why?
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Find the Particular Solution for $y' = 1/x$ if $y(1) = 0$.
The growth rate of the city is $P'(t) = 20e^t + 5$.
Initially, there were 1,000 people.
Task: Find the population formula $P(t)$ and calculate the population after 10 years.
Objective: Explain Initial Conditions using a race.
The Activity:
1. Have two students race.
2. Give one a 5-step head start.
3. Tell them both to walk at the same slow speed.
The Lesson: "Their speed is the same ($f'$), but their position ($f$) is different because of where they started ($C$). God cares about our individual starting lines!"
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