Problem Statement
Solve the second-order differential equation $\frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = 0$ subject to initial conditions $y(0) = 2$ and $y'(0) = 5$.
**(a)** [4 marks] Find the general solution of the differential equation.
**(b)** [3 marks] Apply the initial conditions to determine the particular solution.
Verified Solution & Marking Scheme
Part (a) [M1]
Form auxiliary equation: $r^2 - 6r + 9 = 0$
Part (a) [A1]
Factor: $(r - 3)^2 = 0 \implies r = 3$ (repeated root)
Part (a) [A1]
General solution form: $y(x) = (A + Bx)e^{3x}$
Part (b) [M1]
Apply $y(0) = 2 \implies (A + 0)e^0 = 2 \implies A = 2$
Part (b) [M1]
Differentiate: $y'(x) = B e^{3x} + 3(A + Bx)e^{3x}$. At $x = 0$: $y'(0) = B + 3A = 5$
Part (b) [A1]
$B + 3(2) = 5 \implies B + 6 = 5 \implies B = -1$. Particular solution: $y = (2 - x)e^{3x}$