Challenging +1.8 This is a Further Maths FP2 question requiring systematic differentiation of a second-order DE to find Taylor series coefficients around x=1. While the technique is methodical (differentiate repeatedly, substitute initial conditions), it demands careful algebraic manipulation with product rule applications and tracking multiple derivatives—significantly harder than standard C3/C4 calculus but follows a well-defined algorithm without requiring deep conceptual insight.
4.
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } = x , \quad y = 0 , \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \text { at } x = 1$$
Find a series solution of the differential equation in ascending powers of ( \(x - 1\) ) up to and including the term in \(( x - 1 ) ^ { 3 }\).
4.
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y \frac { \mathrm {~d} y } { \mathrm {~d} x } = x , \quad y = 0 , \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 \text { at } x = 1$$
Find a series solution of the differential equation in ascending powers of ( $x - 1$ ) up to and including the term in $( x - 1 ) ^ { 3 }$.\\
\hfill \mbox{\textit{Edexcel FP2 Q4 [7]}}