4 Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$
Show mark scheme
Show mark scheme source
Question 4:
Answer Marks
Guidance
Answer/Working Mark
Guidance
\(m^2 + 4m + 5 (= 0) \Rightarrow m = \frac{-4 \pm \sqrt{16-20}}{2}\) M1
For attempt to solve correct auxiliary equation
\(= -2 \pm i\) A1
For correct roots
\(CF = e^{-2x}(C\cos x + D\sin x)\) A1\(\sqrt{}\)
For correct CF (here or later). f.t. from \(m\). AEtrig but not forms including \(e^{ix}\)
\(PI = p\sin 2x + q\cos 2x\) B1
For stating a trial PI of the correct form
\(y' = 2p\cos 2x - 2q\sin 2x\), \(y'' = -4p\sin 2x - 4q\cos 2x\) M1
For differentiating PI twice and substituting into the DE
\(\cos 2x(-4q + 8p + 5q) + \sin 2x(-4p - 8q + 5p) = 65\sin 2x\) A1
For correct equation
\(8p + q = 0\), \(p - 8q = 65\), \(p = 1\), \(q = -8\) M1
For equating coefficients of \(\cos 2x\) and \(\sin 2x\) and attempting to solve for \(p\) and/or \(q\)
\(PI = \sin 2x - 8\cos 2x\) A1
For correct \(p\) and \(q\)
\(\Rightarrow y = e^{-2x}(C\cos x + D\sin x) + \sin 2x - 8\cos 2x\) B1\(\sqrt{}\)
For using \(GS = CF + PI\), with 2 arbitrary constants in CF and none in PI
Total: 9 marks
Copy
# Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $m^2 + 4m + 5 (= 0) \Rightarrow m = \frac{-4 \pm \sqrt{16-20}}{2}$ | M1 | For attempt to solve correct auxiliary equation |
| $= -2 \pm i$ | A1 | For correct roots |
| $CF = e^{-2x}(C\cos x + D\sin x)$ | A1$\sqrt{}$ | For correct CF (here or later). f.t. from $m$. **AEtrig** but not forms including $e^{ix}$ |
| $PI = p\sin 2x + q\cos 2x$ | B1 | For stating a trial PI of the correct form |
| $y' = 2p\cos 2x - 2q\sin 2x$, $y'' = -4p\sin 2x - 4q\cos 2x$ | M1 | For differentiating PI twice and substituting into the DE |
| $\cos 2x(-4q + 8p + 5q) + \sin 2x(-4p - 8q + 5p) = 65\sin 2x$ | A1 | For correct equation |
| $8p + q = 0$, $p - 8q = 65$, $p = 1$, $q = -8$ | M1 | For equating coefficients of $\cos 2x$ and $\sin 2x$ and attempting to solve for $p$ and/or $q$ |
| $PI = \sin 2x - 8\cos 2x$ | A1 | For correct $p$ and $q$ |
| $\Rightarrow y = e^{-2x}(C\cos x + D\sin x) + \sin 2x - 8\cos 2x$ | B1$\sqrt{}$ | For using $GS = CF + PI$, with 2 arbitrary constants in CF and none in PI |
**Total: 9 marks**
---
Show LaTeX source
Copy
4 Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$
\hfill \mbox{\textit{OCR FP3 2009 Q4 [9]}}