Challenging +1.2 This is a standard FP2 Taylor series question requiring systematic differentiation and substitution. While it involves multiple derivatives and careful algebraic manipulation, the method is algorithmic: differentiate the given DE twice, verify the given result, then evaluate derivatives at x=0 to build the series. The techniques are well-practiced in FP2, making this moderately above average difficulty but not requiring novel insight.
3.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - y ^ { 2 } , \quad y = 1 \text { at } x = 0 \text {. (I) }$$
(b) By differentiating (I) twice with respect to \(x\), show that
$$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + 2 y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 2 = 0$$
(c) Hence, for (I), find the series solution for \(\boldsymbol { y }\) in ascending powers of \(\boldsymbol { x }\) up to and including the term in \(\boldsymbol { x } ^ { \mathbf { 3 } }\). (4)
3.
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - y ^ { 2 } , \quad y = 1 \text { at } x = 0 \text {. (I) }$$
(b) By differentiating (I) twice with respect to $x$, show that
$$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } + 2 y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 2 = 0$$
(c) Hence, for (I), find the series solution for $\boldsymbol { y }$ in ascending powers of $\boldsymbol { x }$ up to and including the term in $\boldsymbol { x } ^ { \mathbf { 3 } }$. (4)\\
\hfill \mbox{\textit{Edexcel FP2 2003 Q3 [8]}}