Challenging +1.2 This is a standard second-order linear differential equation with constant coefficients from Further Maths FP3. It requires finding the complementary function (complex roots), a particular integral (exponential form), then applying initial conditions to find constants. While systematic and multi-step (10 marks), it follows a well-rehearsed algorithm with no novel insight required, making it moderately above average difficulty.
3 Solve the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 26 \mathrm { e } ^ { x }$$
given that \(y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 11\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).
(10 marks)
3 Solve the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 26 \mathrm { e } ^ { x }$$
given that $y = 5$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 11$ when $x = 0$. Give your answer in the form $y = \mathrm { f } ( x )$.\\
(10 marks)
\hfill \mbox{\textit{AQA FP3 2012 Q3 [10]}}