Moderate -0.3 This is a straightforward separable variables question requiring standard technique: separate variables, integrate both sides (one side needs partial fractions), then apply initial condition. The partial fractions decomposition is simple and all steps are routine for P3 level, making it slightly easier than average but not trivial.
5 The variables \(x\) and \(y\) satisfy the differential equation
$$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( x + 2 )$$
and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\).
Separate variables and obtain \(\int\frac{1}{y}\,dy = \int\frac{x+2}{x+1}\,dx\)
B1
Obtain term \(\ln y\)
B1
Use an appropriate method to integrate \((x+2)/(x+1)\)
M1
*M1
Obtain integral \(x + \ln(x+1)\), or equivalent, e.g. \(\ln(x+1) + x + 1\)
A1
Use \(x = 1\) and \(y = 2\) to evaluate a constant, or as limits
DM1
Obtain correct solution in \(x\) and \(y\) in any form e.g. \(\ln y = x + \ln(x+1) - 1\)
A1
Obtain answer \(y = (x+1)e^{x-1}\)
A1
## Question 5:
| Answer | Mark | Notes |
|--------|------|-------|
| Separate variables and obtain $\int\frac{1}{y}\,dy = \int\frac{x+2}{x+1}\,dx$ | B1 | |
| Obtain term $\ln y$ | B1 | |
| Use an appropriate method to integrate $(x+2)/(x+1)$ | M1 | *M1 |
| Obtain integral $x + \ln(x+1)$, or equivalent, e.g. $\ln(x+1) + x + 1$ | A1 | |
| Use $x = 1$ and $y = 2$ to evaluate a constant, or as limits | DM1 | |
| Obtain correct solution in $x$ and $y$ in any form e.g. $\ln y = x + \ln(x+1) - 1$ | A1 | |
| Obtain answer $y = (x+1)e^{x-1}$ | A1 | |
5 The variables $x$ and $y$ satisfy the differential equation
$$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( x + 2 )$$
and it is given that $y = 2$ when $x = 1$. Solve the differential equation and obtain an expression for $y$ in terms of $x$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q5 [7]}}