What you will learn in Differential Equations

1. 1. What a Differential Equation Says

   An equation about a rate rather than about a value — and why that is how nature is usually described.

2. 2. Slope Fields and Solution Curves

   The picture that comes before any technique — and the click that picks one curve.

3. 3. Order, Degree and Verifying a Solution

   Two classifications that get confused, and a check that is cheaper than solving.

4. 4. Forming a Differential Equation

   Going the other way — eliminating the arbitrary constants from a family of curves.

5. 5. Separable Equations

   Getting all the y on one side and all the x on the other, then integrating both.

6. 6. Homogeneous Equations

   When the equation depends only on y/x, one substitution makes it separable.

7. 7. Linear Equations and the Integrating Factor

   Multiplying by something that turns the left side into a product rule.

8. 8. Growth and Decay Models

   The exponential law derived from its assumption, and where that assumption fails.

9. 9. Newton's Law of Cooling and Mixing Problems

   A rate proportional to a difference, and the equilibrium it approaches without reaching.

10. 10. The Oscillation Equation

    Acceleration proportional to displacement and pointing back — the reason so many unrelated things oscillate identically.

11. 11. Euler's Method

    Stepping along the slope field when there is no formula — and watching the error accumulate.

12. 12. From Situation to Equation to Prediction

    Six steps, one of which is creative and the rest of which can be looked up.