Standard +0.3 This is a straightforward separable variables question requiring standard separation, integration of rational functions, and applying initial conditions. The algebra is clean (separating to get y/(y²+5) dy = dx/(x+1)), and the integrals are routine (½ln(y²+5) and ln(x+1)). Slightly above average difficulty due to the manipulation needed, but still a standard textbook exercise with no novel insight required.
5 The variables \(x\) and \(y\) satisfy the differential equation
$$( x + 1 ) y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 5$$
It is given that \(y = 2\) when \(x = 0\). Solve the differential equation obtaining an expression for \(y ^ { 2 }\) in terms of \(x\).
Separate variables correctly and integrate at least one side
B1
Obtain term \(\ln(x+1)\)
B1
Obtain term of the form \(a\ln(y^2+5)\)
M1
Obtain term \(\frac{1}{2}\ln(y^2+5)\)
A1
Use \(y=2\), \(x=0\) to determine a constant, or as limits, in a solution containing terms \(a\ln(y^2+5)\) and \(b\ln(x+1)\), where \(ab\neq 0\)
M1
Obtain correct solution in any form
A1
Obtain final answer \(y^2 = 9(x+1)^2 - 5\)
A1
Total: 7
## Question 5:
| Answer | Mark | Guidance |
|--------|------|----------|
| Separate variables correctly and integrate at least one side | B1 | |
| Obtain term $\ln(x+1)$ | B1 | |
| Obtain term of the form $a\ln(y^2+5)$ | M1 | |
| Obtain term $\frac{1}{2}\ln(y^2+5)$ | A1 | |
| Use $y=2$, $x=0$ to determine a constant, or as limits, in a solution containing terms $a\ln(y^2+5)$ and $b\ln(x+1)$, where $ab\neq 0$ | M1 | |
| Obtain correct solution in any form | A1 | |
| Obtain final answer $y^2 = 9(x+1)^2 - 5$ | A1 | |
| **Total: 7** | | |
5 The variables $x$ and $y$ satisfy the differential equation
$$( x + 1 ) y \frac { \mathrm {~d} y } { \mathrm {~d} x } = y ^ { 2 } + 5$$
It is given that $y = 2$ when $x = 0$. Solve the differential equation obtaining an expression for $y ^ { 2 }$ in terms of $x$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q5 [7]}}