7 Find the solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 10 \mathrm { e } ^ { 4 x } + 8 \sin 2 x + 4 \cos 2 x$$
given that \(y = 2.5\) when \(x = 0\) and \(y = \frac { \pi } { 4 }\) when \(x = \frac { \pi } { 4 }\).
Show mark scheme
Show mark scheme source
Question 7:
Answer Marks
Guidance
Answer Mark
Guidance
Aux eqn \(m^2+4=0\) M1
PI by correct values of \(m\) seen/used
\((y_{CF}=)\; A\sin 2x + B\cos 2x\) A1
\(A\sin 2x + B\cos 2x\) OE
Try \((y_{PI}=)\; ae^{4x}\) M1
\(+bx\sin 2x + cx\cos 2x\) M1
\((y''_{PI}=)\; 16ae^{4x}+(4b-4cx)\cos 2x-(4c+4bx)\sin 2x\) A1
Correct \((y''_{PI}=)\)
Substitution into \(y''+4y\), dep on 2nd and 3rd M and at least one set of differentiations in form \(ke^{4x}+(p+qx)\cos 2x+(r+sx)\sin 2x\) m1
Non-zero constants \(k,p,q,r,s\)
\(20a=10 \Rightarrow a=0.5\); \(-4c=8 \Rightarrow c=-2\); \(4b=4 \Rightarrow b=1\) \((y_{PI}=)\; 0.5e^{4x}\) B1
\(0.5e^{4x}\) term in PI
\(+x\sin 2x - 2x\cos 2x\) B1
\(+x\sin 2x - 2x\cos 2x\) term in PI with correct \(x\sin 2x\) and \(x\cos 2x\) terms in m1 line
\(y=2.5, x=0\): \(B+0.5=2.5 \Rightarrow B=2\) \(y=\frac{\pi}{4}, x=\frac{\pi}{4}\): \(A+\frac{e^\pi}{2}+\frac{\pi}{4}=\frac{\pi}{4}\); \(A=-\frac{e^\pi}{2}\) A1F
Correct ft value of either \(A\) or \(B\); coeffs of \(\sin 2x\) and \(\cos 2x\) respectively; m1 must have been scored
\(y=2(1-x)\cos 2x + x\sin 2x - \frac{e^\pi}{2}\sin 2x + \frac{e^{4x}}{2}\) A1
ACF. Must be correct eqn and ALL previous 9 marks scored
Copy
# Question 7:
| Answer | Mark | Guidance |
|--------|------|----------|
| Aux eqn $m^2+4=0$ | M1 | PI by correct values of $m$ seen/used |
| $(y_{CF}=)\; A\sin 2x + B\cos 2x$ | A1 | $A\sin 2x + B\cos 2x$ OE |
| Try $(y_{PI}=)\; ae^{4x}$ | M1 | |
| $+bx\sin 2x + cx\cos 2x$ | M1 | |
| $(y''_{PI}=)\; 16ae^{4x}+(4b-4cx)\cos 2x-(4c+4bx)\sin 2x$ | A1 | Correct $(y''_{PI}=)$ |
| Substitution into $y''+4y$, dep on 2nd and 3rd M and at least one set of differentiations in form $ke^{4x}+(p+qx)\cos 2x+(r+sx)\sin 2x$ | m1 | Non-zero constants $k,p,q,r,s$ |
| $20a=10 \Rightarrow a=0.5$; $-4c=8 \Rightarrow c=-2$; $4b=4 \Rightarrow b=1$ | | |
| $(y_{PI}=)\; 0.5e^{4x}$ | B1 | $0.5e^{4x}$ term in PI |
| $+x\sin 2x - 2x\cos 2x$ | B1 | $+x\sin 2x - 2x\cos 2x$ term in PI with correct $x\sin 2x$ and $x\cos 2x$ terms in m1 line |
| $y=2.5, x=0$: $B+0.5=2.5 \Rightarrow B=2$ | | |
| $y=\frac{\pi}{4}, x=\frac{\pi}{4}$: $A+\frac{e^\pi}{2}+\frac{\pi}{4}=\frac{\pi}{4}$; $A=-\frac{e^\pi}{2}$ | A1F | Correct ft value of either $A$ or $B$; coeffs of $\sin 2x$ and $\cos 2x$ respectively; m1 must have been scored |
| $y=2(1-x)\cos 2x + x\sin 2x - \frac{e^\pi}{2}\sin 2x + \frac{e^{4x}}{2}$ | A1 | ACF. Must be correct eqn and ALL previous 9 marks scored |
---
Show LaTeX source
Copy
7 Find the solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 10 \mathrm { e } ^ { 4 x } + 8 \sin 2 x + 4 \cos 2 x$$
given that $y = 2.5$ when $x = 0$ and $y = \frac { \pi } { 4 }$ when $x = \frac { \pi } { 4 }$.\\[0pt]
[10 marks]
\hfill \mbox{\textit{AQA FP3 2016 Q7 [10]}}