| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Topic | Second order differential equations |
| Type | Standard non-homogeneous with exponential RHS |
| Difficulty | Standard +0.8 This is a standard second-order linear non-homogeneous differential equation requiring the complementary function (solving the auxiliary equation with complex roots) and a particular integral (using the exponential substitution method). While methodical, it involves multiple techniques and careful algebraic manipulation, placing it moderately above average difficulty for A-level, though routine for Further Maths students. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
**Question 5** [8 marks]
Aux. Eqn. $m^2-4m+5=0$ M1 (Including solving attempt)
$m = 2 \pm i$ A1
Comp. Fn. is $y_C = e^{2x}(A\cos x + B\sin x)$ B1ft
For Part. Intgl. try $y = y_p = ae^{2x}$ B1
Both $y' = 2ae^{2x}$ and $y'' = 4ae^{2x}$ B1
Subst. into given d.e. & solving to find $a$: M1 $(4a-8a+5a)e^{2x} = 24e^{2x}$
$y_p = 24e^{2x}$ A1
Gen. Soln. $y = e^{2x}(A\cos x + B\sin x + 24)$ B1ft ($y_C + y_p$ provided $y_C$ has 2 arbitrary constants and $y_p$ has none. Also, $A$, $B$ must be real here)
**[8]**5 Find the general solution of the differential equation $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 24 \mathrm { e } ^ { 2 x }$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2016 Q5 [8]}}