Standard +0.8 This is a standard integrating factor question from Further Maths FP2, requiring identification of the linear form, finding integrating factor sec²x, and integrating e^(4x)sec²x which needs integration by parts twice. The particular solution adds minimal difficulty. While methodical, the integration is non-trivial and this is Further Maths content, placing it moderately above average difficulty.
3. (a) Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \mathrm { e } ^ { 4 x } \cos ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
giving your answer in the form \(y = \mathrm { f } ( x )\).
(b) Find the particular solution for which \(y = 1\) at \(x = 0\)
Using given initial conditions to obtain value for \(c\)
\(c = \frac{3}{4}\)
\(y = \frac{1}{4}(e^{4x}+3)\cos^2 x\)
A1
Fully correct final answer; may be in form \(y\sec^2 x = \ldots\) or \(4y\sec^2 x = \ldots\)
# Question 3:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $e^{2\int \tan x\, dx} = e^{2\ln\sec x} = \sec^2 x$ or $\frac{1}{\cos^2 x}$ | M1A1 | M1: attempting integrating factor, including integration of $2\tan x$, $\ln\cos$ or $\ln\sec$ seen; A1: correct IF |
| $\sec^2 x \frac{dy}{dx} + 2y\tan x \sec^2 x = e^{4x}\cos^2 x \sec^2 x$ | dM1 | Multiplying equation by IF (may be implied by next line) |
| $\frac{d}{dx}(y\sec^2 x) = e^{4x}$ | B1ft | $y \times$ their IF |
| $y\sec^2 x = \frac{1}{4}e^{4x}$ $(+c)$ | M1 | Attempting complete integration of RHS; must include $ke^{4x}$; constant needed |
| $y = \left(\frac{1}{4}e^{4x} + c\right)\cos^2 x$ | A1 | Correct solution, constant must be included |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y=1, x=0$: $1 = \left(\frac{1}{4} + c\right)$ | M1 | Using given initial conditions to obtain value for $c$ |
| $c = \frac{3}{4}$ | | |
| $y = \frac{1}{4}(e^{4x}+3)\cos^2 x$ | A1 | Fully correct final answer; may be in form $y\sec^2 x = \ldots$ or $4y\sec^2 x = \ldots$ |
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3. (a) Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \tan x = \mathrm { e } ^ { 4 x } \cos ^ { 2 } x , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$
giving your answer in the form $y = \mathrm { f } ( x )$.\\
(b) Find the particular solution for which $y = 1$ at $x = 0$\\
\hfill \mbox{\textit{Edexcel FP2 2014 Q3 [8]}}