Standard +0.3 This is a standard integrating factor question from Further Maths with straightforward application of the method. The integrating factor is x^(-2), leading to integration by parts for x·e^(2x), which is routine for FP3 students. While it requires multiple steps and careful algebra, it follows a well-practiced algorithm with no conceptual surprises.
4 By using an integrating factor, find the solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = 2 x ^ { 3 } \mathrm { e } ^ { 2 x }$$
given that \(y = \mathrm { e } ^ { 4 }\) when \(x = 2\). Give your answer in the form \(y = \mathrm { f } ( x )\).
4 By using an integrating factor, find the solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = 2 x ^ { 3 } \mathrm { e } ^ { 2 x }$$
given that $y = \mathrm { e } ^ { 4 }$ when $x = 2$. Give your answer in the form $y = \mathrm { f } ( x )$.
\hfill \mbox{\textit{AQA FP3 2011 Q4 [9]}}