Find the general solution of (x + 2y³)(dy/dx) = y, y > 0.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Write what is given and what is to be found. Given: (x + 2y³)(dy/dx) = y with y > 0. To find: the general solution.
2 · Decide the type. dy/dx = y/(x + 2y³) is not linear in y. Flip it: dx/dy = (x + 2y³)/y = x/y + 2y².
3 · Write the standard form and the formula. dx/dy − (1/y)x = 2y², so P₁ = −1/y and Q₁ = 2y². Solution: x × IF = ∫(Q₁ × IF) dy + C.
4 · Find the integrating factor. IF = e^(∫ −(1/y) dy) = e^(−log y) = 1/y.
5 · Work out the solution.
- x × (1/y) = ∫ 2y² × (1/y) dy + C
- x/y = ∫ 2y dy + C = y² + C
6 · Check and box the answer. x = y³ + Cy. Check: dx/dy = 3y² + C, and x/y + 2y² = y² + C + 2y² = 3y² + C.

Scoring tip: Say in words why you flipped the equation ('not linear in y, linear in x'). It shows the examiner you chose the method on purpose.