| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.3 This is a straightforward separable variables question requiring standard techniques: separate to get e^(-y)dy = xe^x dx, integrate both sides (RHS needs integration by parts), apply initial condition, and rearrange for y. Part (ii) requires recognizing the domain restriction from ln(1-e^x), which is a simple observation once the solution is obtained. Slightly above average due to the integration by parts and final rearrangement, but still a standard textbook exercise. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Separate variables and attempt integration of one side | M1 | |
| Obtain term \(-e^{-y}\) | A1 | |
| Integrate \(xe^x\) by parts reaching \(xe^{\pm}\int e^x \, dx\) | M1 | |
| Obtain integral \(xe^x - e^x\) | A1 | |
| Evaluate a constant, or use limits \(x = 0, y = 0\) | M1 | |
| Obtain correct solution in any form | A1 | |
| Obtain final answer \(y = -\ln(e^x(1-x))\), or equivalent | A1 | [7] |
| (ii) Justify the given statement | B1 | [1] |
(i) Separate variables and attempt integration of one side | M1 |
Obtain term $-e^{-y}$ | A1 |
Integrate $xe^x$ by parts reaching $xe^{\pm}\int e^x \, dx$ | M1 |
Obtain integral $xe^x - e^x$ | A1 |
Evaluate a constant, or use limits $x = 0, y = 0$ | M1 |
Obtain correct solution in any form | A1 |
Obtain final answer $y = -\ln(e^x(1-x))$, or equivalent | A1 | [7]
(ii) Justify the given statement | B1 | [1]7 The variables $x$ and $y$ satisfy the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { x + y }$$
and it is given that $y = 0$ when $x = 0$.\\
(i) Solve the differential equation and obtain an expression for $y$ in terms of $x$.\\
(ii) Explain briefly why $x$ can only take values less than 1 .
\hfill \mbox{\textit{CAIE P3 2016 Q7 [8]}}