Standard +0.8 This question requires systematic differentiation of a differential equation to find successive derivatives at x=0, then construct a Taylor series. While the technique is standard for FP2, it demands careful algebraic manipulation through multiple derivatives (up to 5th order) and correct application of the Taylor series formula. The method is taught explicitly but requires sustained accuracy across several steps, making it moderately challenging but not requiring novel insight.
$$\frac{d^2 y}{dx^2} + 4y - \sin x = 0$$
Given that \(y = \frac{1}{2}\) and \(\frac{dy}{dx} = \frac{1}{8}\) at \(x = 0\),
find a series expansion for \(y\) in terms of \(x\), up to and including the term in \(x^5\). [5]
$$\frac{d^2 y}{dx^2} + 4y - \sin x = 0$$
Given that $y = \frac{1}{2}$ and $\frac{dy}{dx} = \frac{1}{8}$ at $x = 0$,
find a series expansion for $y$ in terms of $x$, up to and including the term in $x^5$. [5]
\hfill \mbox{\textit{Edexcel FP2 Q3 [5]}}