Challenging +1.8 This is a challenging Further Maths question requiring multiple sophisticated techniques: deriving first and second derivatives under substitution y=1/w, algebraic manipulation to verify the reduction, solving a second-order non-homogeneous linear ODE with constant coefficients, and asymptotic analysis. While the individual steps are systematic, the combination of substitution verification, solving the ODE, and finding the asymptotic behavior requires strong technical facility and multi-step reasoning beyond typical A-level questions.
Show that the substitution \(y = \frac { 1 } { w }\) reduces the differential equation
$$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 5 y ^ { 2 } = \left( 5 x ^ { 2 } + 4 x + 2 \right) y ^ { 3 }$$
to
$$\frac { \mathrm { d } ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} w } { \mathrm {~d} x } + 5 w = - 5 x ^ { 2 } - 4 x - 2$$
Find the general solution for \(w\) in terms of \(x\).
Find a function f such that \(\lim _ { x \rightarrow \infty } \left( \frac { y } { \mathrm { f } ( x ) } \right) = 1\).
Show that the substitution $y = \frac { 1 } { w }$ reduces the differential equation
$$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 5 y ^ { 2 } = \left( 5 x ^ { 2 } + 4 x + 2 \right) y ^ { 3 }$$
to
$$\frac { \mathrm { d } ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} w } { \mathrm {~d} x } + 5 w = - 5 x ^ { 2 } - 4 x - 2$$
Find the general solution for $w$ in terms of $x$.
Find a function f such that $\lim _ { x \rightarrow \infty } \left( \frac { y } { \mathrm { f } ( x ) } \right) = 1$.
\hfill \mbox{\textit{CAIE FP1 2007 Q12 OR}}