Challenging +1.2 This is a standard Further Maths FP2 technique: using Taylor series to solve a differential equation with initial conditions. Students differentiate the DE repeatedly, substitute initial conditions, and build the series term-by-term. While it requires careful algebraic manipulation and understanding of the small-angle approximation (sin x ≈ x), it's a well-practiced exam technique with a clear algorithmic procedure. The question is moderately harder than average A-level due to being Further Maths content, but it's routine within FP2.
7.
$$\frac { \mathrm { d } ^ { 2 x } } { \mathrm {~d} t ^ { 2 } } + 3 \sin x = 0 . \quad \text { At } t = 0 , \quad x = 0 \quad \text { and } \quad \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4$$
(b) Find a series solution for \(x\), in ascending powers of \(t\), up to and including the term in \(t ^ { 3 }\).
(c) Use your answer to (b) to obtain an estimate of \(x\) at \(t = 0.3\).
(2)(Total 11 marks)
7.
$$\frac { \mathrm { d } ^ { 2 x } } { \mathrm {~d} t ^ { 2 } } + 3 \sin x = 0 . \quad \text { At } t = 0 , \quad x = 0 \quad \text { and } \quad \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4$$
(b) Find a series solution for $x$, in ascending powers of $t$, up to and including the term in $t ^ { 3 }$.\\
(c) Use your answer to (b) to obtain an estimate of $x$ at $t = 0.3$.\\
(2)(Total 11 marks)\\
\hfill \mbox{\textit{Edexcel FP2 2006 Q7 [11]}}