Intro to Differential Equations
A Differential Equation (DE) is an equation containing a derivative, like y' = 2x or dy/dx = x + y.
To check if y = f(x) is a solution to a DE:
- Find y' (and y'' if needed) from the given solution.
- Plug y and y' into the DE.
- If LHS = RHS, it is a valid solution.
Verify that y = e^{3x} is a solution to y' - 3y = 0.
- Derive: y = e^{3x} → y' = 3e^{3x}.
- Plug in: (3e^{3x}) - 3(e^{3x}).
- Simplify: 0 = 0. (Verified!)
Slope Fields
A Slope Field is a graphical representation of a Differential Equation. At every point (x, y), we draw a tiny line segment with slope dy/dx.
Drawing Them
Plug the coordinates (x, y) into the DE to calculate the slope. Draw a short line with that slope.
Reading Them
Follow the "flow" of the lines. Solutions follow the path of the segments, like a leaf in a stream.
- Zero Slope Check: Look where the slopes are horizontal (m=0). If the DE is dy/dx = x+y, slopes must be 0 along the line y = -x.
- Undefined Slope Check: Look for vertical lines. If the DE is dy/dx = -x/y, slopes are vertical when y=0 (x-axis).
- Variable Check: If DE depends only on x, slopes in columns are identical. If only on y, slopes in rows are identical.
Separation of Variables
This is the ONLY algebraic method you need to know to solve DEs in AB Calculus. Mastering this is usually worth 5-6 points on an FRQ.
- Separate: Move all y terms to the dy side and x terms to the dx side. (Multiply/Divide only!).
- Integrate: Add ∫ to both sides.
- + C: Add + C immediately to the x side.
- Evaluate C: Use the Initial Condition (x_0, y_0) to solve for C.
- Solve for y: Isolate y to get the Particular Solution y = f(x).
Solve: dy/dx = xy² with y(0) = 1.
- Separate: 1/y² dy = x dx
- Integrate: ∫ y⁻² dy = ∫ x dx → -1/y = x²/2 + C
- Find C: Plug in (0, 1): -1/1 = 0 + C → C = -1.
- Refine: -1/y = x²/2 - 1
- Solve y: y = -1 / (x²/2 - 1)
Exponential Growth & Decay
When the rate of change is proportional to the amount present, we get the classic exponential growth model.
Solution:
Population, Bacteria, Interest.
Radioactive Half-Life, Cooling (Newton's Law).