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 general solution describes every solution of a differential equation, using one arbitrary constant.
- If dy/dx = f(x) only, then y = ∫ f(x) dx: any antiderivative plus C.
- Separable: dy/dx = g(x) · h(y). Try factoring (xy + 2x = x(y + 2)) and exponent laws (e^(x + y) = eˣeʸ) before deciding.
- Not separable: sums such as x + y, or cos(x + y).
- Method: (1/h(y)) dy = g(x) dx → integrate both sides → add one C → solve for y (or leave implicit) → check by differentiating.
- The chain rule justifies the method: d/dx [H(y)] = H′(y) · dy/dx.
Key relationships
| Situation | What to write |
|---|---|
| ln|y| = G(x) + C | |y| = e^C e^(G(x)), so y = A e^(G(x)) with A any real constant |
| y³ = (expression) + C | y = ∛(expression + C), no sign choice needed |
| eʸ = (expression) + C | y = ln(expression + C), domain needs expression + C > 0 |
| y cannot be isolated | Leave it implicit, e.g. sin y + y³ = x² + C |
| ∫ (1/y²) dy | −1/y, not a logarithm |
| ∫ (1/(1 + y²)) dy | arctan y |
Assumptions behind the method
- You divide by h(y), so you assume h(y) ≠ 0. Constant solutions where h(y) = 0 (such as y = 0) must be checked separately; often A = 0 covers them.
- Each side is integrated with respect to its own variable only.
Mistakes to avoid
- No constant (one solution, not the general solution).
- Constant added after rearranging: e^(G(x)) + C instead of A e^(G(x)).
- Logarithm for every fraction.
- “Integrating” y with respect to x without separating, as in ∫ (x + y) dx = x²/2 + xy.
- Two constants where one will do.
Quick self-check
- Find the general solution of dy/dx = 2x/y. (y dy = 2x dx, so y²/2 = x² + C, or y² = 2x² + K)
- Find the general solution of dy/dx = y sec²x. (ln|y| = tan x + C, so y = A e^(tan x))
- Is dy/dx = x + y² separable? (No. It is a sum, and it cannot be factored into an x-part times a y-part.)
Next: practice questions.