Verify that y = Ae²ˣ + Be⁻ˣ is a solution of d²y/dx² − dy/dx − 2y = 0. State whether it is a general or a particular solution.

1. 1 · Write what is given and what is to be found. Given: y = Ae²ˣ + Be⁻ˣ and the DE y″ − y′ − 2y = 0. To show: y satisfies the DE, and to say which type of solution it is.
2. 2 · Write the rule. Differentiate as many times as the order (here 2), substitute and show LHS = RHS.
3. 3 · Find the derivatives.
   • y′ = 2Ae²ˣ − Be⁻ˣ
   • y″ = 4Ae²ˣ + Be⁻ˣ
4. 4 · Substitute and simplify.
   • LHS = (4Ae²ˣ + Be⁻ˣ) − (2Ae²ˣ − Be⁻ˣ) − 2(Ae²ˣ + Be⁻ˣ)
   • = (4A − 2A − 2A)e²ˣ + (B + B − 2B)e⁻ˣ = 0 = RHS
5. 5 · Classify the solution. The DE has order 2 and the solution has two arbitrary constants A and B, so it is the general solution.
6. 6 · Check and box the answer. y = Ae²ˣ + Be⁻ˣ satisfies the DE; it is the general solution.

Scoring tip: Collect the e²ˣ terms and the e⁻ˣ terms separately. Each bracket becoming 0 is easy for the examiner to see.