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University Math Lab: Systems & Structures FREE
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Differential Equations & Slope Fields
Learning Objectives
- Sketch slope fields
- Solve separable ODEs
- Apply initial conditions
Key Concepts
Slope Field
Grid of line segments showing dy/dx at each point
Separable ODE
dy/dx = f(x)g(y) — can separate variables
Theory
**Slope field:** At each point (x,y), draw a small line with slope dy/dx = f(x,y).
**Separable equations:** dy/dx = f(x)g(y)
1. Separate: dy/g(y) = f(x)dx
2. Integrate both sides
3. Apply initial condition
**Example:** dy/dx = xy
- dy/y = x dx
- ln|y| = x²/2 + C
- y = Ae^(x²/2)
Examples
Solve dy/dx = 2x, y(0) = 3
Solution: y = x² + C. y(0)=3 → C=3. y = x² + 3