Short on time? This page is the recap. For explanations, the family-of-curves graph and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- A solution of a differential equation is a function that makes both sides equal for every input in an interval.
- Method: (1) find y′ (and y″ if needed); (2) substitute into the left side and the right side separately; (3) simplify each side; (4) show they are identical.
- Agreement at one x value does not prove a solution. Disagreement at one x value does disprove it.
- A differential equation can have infinitely many solutions, often written with an arbitrary constant C (or two constants for a second-order equation).
- A constant function y = c is a solution when substituting y = c makes the right side 0 for every x (its derivative is 0).
- A solution must be differentiable on its whole interval, so avoid points where the formula is undefined.
Key relationships
| Differential equation | Family of solutions | Check |
|---|---|---|
| dy/dx = ky | y = Ce^(kx) | y′ = kCe^(kx) = ky |
| dy/dt = k(M − y) | y = M + Ce^(−kt) | y′ = −kCe^(−kt); k(M − y) = −kCe^(−kt) |
| y″ + ω²y = 0 (ω a positive constant) | y = A sin(ωx) + B cos(ωx) | y″ = −ω²y |
| ay″ + by′ + cy = 0 | try y = e^(rx) | needs ar² + br + c = 0 |
To find a constant that makes a form work, substitute and match coefficients so the equation holds for all x.
Assumptions
- C, A, B and k are constants: differentiate them as constants.
- Differentiate with respect to the variable the equation uses (x or t).
- State the interval when the formula has points where it is undefined.
Mistakes to avoid
- Checking at one point and calling it a proof.
- Working across the equals sign as if the equation were already true.
- Missing the chain-rule factor, for example in e^(−3t) or sin(3x).
- Forgetting y″ in a second-order equation.
- Adding a constant to a solution and assuming the result is still a solution.
- Saying “the” solution when there is a whole family.
Quick self-check
- Is y = 7e^(−2x) a solution of dy/dx = −2y? (Yes: both sides are −14e^(−2x).)
- Is y = x² a solution of dy/dx = 2y/x for x > 0? (Yes: both sides are 2x.)
- Find r so that y = e^(rx) solves y″ = 9y. (r² = 9, so r = 3 or r = −3.)
- Both sides of a differential equation equal 5 at x = 1. Is the function a solution? (Not proven; you must show equality for all x.)
Next: practice questions.