Standard +0.3 This is a standard second-order linear differential equation with constant coefficients and polynomial forcing term. Students follow a routine procedure: find complementary function (complex roots requiring exponential-trigonometric form), find particular integral (polynomial trial solution with undetermined coefficients), then apply initial conditions. While it requires multiple steps and careful algebra, it's a textbook exercise with no novel insight required, making it slightly easier than average for Further Maths.
Find the particular solution of the differential equation
$$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 3y = 27x^2,$$
given that, when \(x = 0\), \(y = 2\) and \(\frac{dy}{dx} = -8\). [10]
Find the particular solution of the differential equation
$$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 3y = 27x^2,$$
given that, when $x = 0$, $y = 2$ and $\frac{dy}{dx} = -8$. [10]
\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q4 [10]}}