What you'll learn
Equations involving a derivative, and how to model change with them.
Slope fields
A grid of short segments with slope dy/dx at each point; a solution curve follows the segments. Use them to sketch solutions and read behavior without solving.
Separable equations
If dy/dx = g(x)h(y), separate variables: (1/h(y)) dy = g(x) dx, integrate both sides (+C), then apply an initial condition to find C.
- Example: dy/dx = ky → dy/y = k dx → ln|y| = kx + C → y = A·eᵏˣ (exponential growth/decay).
Exponential models
dy/dt = ky means the rate is proportional to the amount → y = y₀eᵏᵗ. k > 0 growth, k < 0 decay.
Exam tips
- Always apply the initial condition to solve for the constant.
- Match a slope field to its differential equation by checking slopes at a few points.
Common mistakes
- Forgetting + C before applying the initial condition.
- Algebra errors exponentiating to remove the logarithm.