Challenging +1.2 This is a Further Maths FP3 question on second-order differential equations with resonance. While the topic itself is advanced (Further Maths), the question provides the complementary function and students only need to find a particular integral using the modified form y_p = Cx²e^x due to resonance, then combine solutions. It's a standard textbook exercise for this module requiring methodical application of a known technique rather than novel insight.
3 It is given that the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$
is \(y = \mathrm { e } ^ { x } ( A x + B )\). Hence find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 6 \mathrm { e } ^ { x }$$
3 It is given that the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$
is $y = \mathrm { e } ^ { x } ( A x + B )$. Hence find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 6 \mathrm { e } ^ { x }$$
\hfill \mbox{\textit{AQA FP3 2013 Q3 [5]}}