Challenging +1.2 This is a standard Further Maths technique requiring successive differentiation of the differential equation to find Taylor coefficients. While it involves multiple steps and careful algebraic manipulation, it follows a well-established algorithmic procedure taught explicitly in FP2. The initial conditions are given directly, making this a methodical rather than insightful problem—harder than average A-level but routine for Further Maths students.
2. The displacement \(x\) metres of a particle at time \(t\) seconds is given by the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + x + \cos x = 0$$
When \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }\).
Find a Taylor series solution for \(x\) in ascending powers of \(t\), up to and including the term in \(t ^ { 3 }\).
2. The displacement $x$ metres of a particle at time $t$ seconds is given by the differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + x + \cos x = 0$$
When $t = 0 , x = 0$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 1 } { 2 }$.\\
Find a Taylor series solution for $x$ in ascending powers of $t$, up to and including the term in $t ^ { 3 }$.\\
\hfill \mbox{\textit{Edexcel FP2 2010 Q2 [5]}}