Why this chapter matters
Differential equations describe systems where change depends on current state, including growth, cooling, and mixing behavior.
What you will learn
- Interpret slope fields to compare families of possible solutions.
- Solve separable first-order differential equations.
- Use initial conditions to determine a specific solution curve.
Lessons in this chapter
- From verbal model to differential equationTranslate rate statements into equations involving derivatives.
- Slope fields and qualitative behaviorUse local slope patterns to predict long-run tendencies.
- Separable equationsSeparate variables, integrate both sides, and apply constants correctly.
- Exponential growth and decay solutionsUse dy/dt = ky with initial value data to model changing quantities.
Study task
Solve dy/dx = 2xy with y(0) = 3 and explain how the initial condition changes a family of solutions into one function.
Chapter checkpoint
Solve dy/dx = 4x with y(1) = 5.
Integrate to get y = 2x^2 + C. Use y(1) = 5, so 2 + C = 5 and C = 3. The solution is y = 2x^2 + 3.