| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Standard +0.8 This is a separable differential equation requiring algebraic manipulation to separate variables, integration of a rational function requiring partial fractions, application of initial conditions, and rearrangement to make y the subject. Part (b) requires limit analysis. While the techniques are standard A-level, the multi-step nature, algebraic complexity, and need to handle the rational function correctly make this moderately challenging—above average but not exceptionally difficult. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct separation of variables | B1 | \(\displaystyle\int\dfrac{1}{y^2+y}\,dy = \int-\dfrac{1}{x^2}\,dx\). Condone missing integral signs or missing \(dx\), \(dy\), but not both. |
| Obtain \(\dfrac{1}{x}\) | B1 | |
| Express \(\dfrac{1}{y^2+y}\) in partial fractions or express denominator as difference of two squares | *M1 | Allow for the correct split of \(\dfrac{\pm1}{(y^2\pm y)}\) |
| Obtain \(\dfrac{1}{y}-\dfrac{1}{y+1}\) or \(\dfrac{1}{\left(y+\frac{1}{2}\right)^2-\left(\frac{1}{2}\right)^2}\) | A1 | Allow if coefficients for the partial fractions are correct but followed by an error |
| Obtain \(\ln y - \ln(y+1)\) | A1 | Or equivalent, dependent on where they left the minus sign |
| Use \(x=1\), \(y=1\) to find constant of integration or as limits in a definite integral in an expression containing terms of the form \(\frac{p}{x}\), \(q\ln y\) and \(r\ln(1+y)\) | DM1 | \(\ln\frac{1}{2}=1+C\). If they rearrange the equation before finding the constant of integration then the constant must be of the correct form. |
| Correct equation in \(x\) and \(y\) | A1 | \(\ln\dfrac{y}{1+y}=\dfrac{1}{x}-1+\ln\dfrac{1}{2}\) |
| Obtain \(y=\dfrac{e^{\frac{1}{x}-1}}{2-e^{\frac{1}{x}-1}}\) | A1 | Or equivalent e.g. \(y=\dfrac{1}{2e^{1-\frac{1}{x}}-1}\), \(y=\dfrac{1}{e^{1-\frac{1}{x}+\ln 2}-1}\). Accept with decimal value for \(e^{-1}\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State that \(y\) approaches \(\dfrac{1}{2e-1}\) | B1 FT | Or exact equivalent. Condone \(y=\dfrac{1}{2e-1}\). FT on an expression in \(e^{\frac{1}{x}}\). |
## Question 11(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct separation of variables | B1 | $\displaystyle\int\dfrac{1}{y^2+y}\,dy = \int-\dfrac{1}{x^2}\,dx$. Condone missing integral signs or missing $dx$, $dy$, but not both. |
| Obtain $\dfrac{1}{x}$ | B1 | |
| Express $\dfrac{1}{y^2+y}$ in partial fractions or express denominator as difference of two squares | *M1 | Allow for the correct split of $\dfrac{\pm1}{(y^2\pm y)}$ |
| Obtain $\dfrac{1}{y}-\dfrac{1}{y+1}$ or $\dfrac{1}{\left(y+\frac{1}{2}\right)^2-\left(\frac{1}{2}\right)^2}$ | A1 | Allow if coefficients for the partial fractions are correct but followed by an error |
| Obtain $\ln y - \ln(y+1)$ | A1 | Or equivalent, dependent on where they left the minus sign |
| Use $x=1$, $y=1$ to find constant of integration or as limits in a definite integral in an expression containing terms of the form $\frac{p}{x}$, $q\ln y$ and $r\ln(1+y)$ | DM1 | $\ln\frac{1}{2}=1+C$. If they rearrange the equation before finding the constant of integration then the constant must be of the correct form. |
| Correct equation in $x$ and $y$ | A1 | $\ln\dfrac{y}{1+y}=\dfrac{1}{x}-1+\ln\dfrac{1}{2}$ |
| Obtain $y=\dfrac{e^{\frac{1}{x}-1}}{2-e^{\frac{1}{x}-1}}$ | A1 | Or equivalent e.g. $y=\dfrac{1}{2e^{1-\frac{1}{x}}-1}$, $y=\dfrac{1}{e^{1-\frac{1}{x}+\ln 2}-1}$. Accept with decimal value for $e^{-1}$. |
**Total: 8 marks**
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## Question 11(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State that $y$ approaches $\dfrac{1}{2e-1}$ | B1 FT | Or exact equivalent. Condone $y=\dfrac{1}{2e-1}$. FT on an expression in $e^{\frac{1}{x}}$. |
**Total: 1 mark**11 The variables $x$ and $y$ satisfy the differential equation
$$x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y ^ { 2 } + y = 0$$
It is given that $x = 1$ when $y = 1$.
\begin{enumerate}[label=(\alph*)]
\item Solve the differential equation to obtain an expression for $y$ in terms of $x$.
\item State what happens to the value of $y$ when $x$ tends to infinity. Give your answer in an exact form.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q11 [9]}}