Standard +0.8 This is a standard first-order linear ODE requiring the integrating factor method, but the coefficient (4t²-1) makes it more involved than typical A-level examples. Students must rearrange to standard form, find the integrating factor involving ln|4t²-1|, integrate products carefully, and apply the initial condition. The algebraic manipulation is non-trivial but follows a well-defined procedure for Further Maths students.
4 Find the solution of the differential equation
$$\left( 4 t ^ { 2 } - 1 \right) \frac { d x } { d t } + 4 x = 4 t ^ { 2 } - 1$$
for which \(x = 3\) when \(t = 1\). Give your answer in the form \(\mathrm { x } = \mathrm { f } ( \mathrm { t } )\).
After integrating, divides their expression through by their coefficient of \(x\) (their coefficient must depend on \(t\))
9
## Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{dx}{dt}+\frac{4}{4t^2-1}x=1$ | **B1** | Divides through by $4t^2-1$ |
| $\frac{4}{4t^2-1}=\frac{2}{2t-1}-\frac{2}{2t+1}$ or $\frac{4}{4t^2-1}=\frac{4}{(2t)^2-1}=\frac{1}{t^2-(\frac{1}{2})^2}$ | **B1** | Writes as partial fractions or a form to which the formula from MF19 can be applied |
| $e^{\int 4(4t^2-1)^{-1}\,dt}=e^{\ln(2t-1)-\ln(2t+1)}=\frac{2t-1}{2t+1}$ | **M1 A1** | Finds integrating factor |
| $\frac{d}{dt}\!\left(x\frac{2t-1}{2t+1}\right)=\frac{2t-1}{2t+1}=\frac{2t+1-2}{2t+1}$ | **M1** | Correct form on LHS and attempt to integrate RHS. RHS must be of the form $\frac{at+b}{ct+d}$ with $a,b,c,d\neq 0$ |
| $x\frac{2t-1}{2t+1}=t-\ln(2t+1)+C$ | **A1** | |
| $1=1-\ln 3+C$ | **M1** | Substitutes into their expression in $x$ and $t$ |
| $x=\left(\frac{2t+1}{2t-1}\right)\!\left(t+\ln\!\left(\frac{3}{2t+1}\right)\right)$ | **M1 A1** | After integrating, divides their expression through by their coefficient of $x$ (their coefficient must depend on $t$) |
| | **9** | |
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4 Find the solution of the differential equation
$$\left( 4 t ^ { 2 } - 1 \right) \frac { d x } { d t } + 4 x = 4 t ^ { 2 } - 1$$
for which $x = 3$ when $t = 1$. Give your answer in the form $\mathrm { x } = \mathrm { f } ( \mathrm { t } )$.\\
\hfill \mbox{\textit{CAIE Further Paper 2 2022 Q4 [9]}}