Given that the general solution of the differential equation (x + 1) dy/dx = y(x^2 - 1) is y = A(x - 1)e^(x^2/2), where A is an arbitrary constant. (a) Show that y = A(x - 1)e^(x^2/2) satisfies the differential equation. (b) Find the particular solution of this differential equation for which y = 3 when x = 2.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1: Uses the product rule to differentiate y = A(x - 1)e^(x^2/2). dy/dx = A(1)e^(x^2/2) + A(x - 1)(x)e^(x^2/2).
2. A1: Correct dy/dx in an unsimplified form.
3. M1: Substitutes their dy/dx into the LHS of the differential equation: (x + 1) [A(1)e^(x^2/2) + A(x - 1)(x)e^(x^2/2)].
4. M1: Substitutes y into the RHS of the differential equation: A(x - 1)e^(x^2/2) (x^2 - 1).
5. A1: Manipulates LHS to show it is equal to RHS (or vice versa), e.g., (x + 1)A e^(x^2/2) [1 + x(x - 1)] = A e^(x^2/2) (x - 1)(x + 1)(x - 1).
6. A1: Clearly shows that LHS = RHS, thus satisfying the differential equation.
7. M1: Substitutes x = 2 and y = 3 into the general solution y = A(x - 1)e^(x^2/2).
8. A1: Correct value for A (e.g., A = 3/e^2) and writes down the particular solution.