Moderate -0.3 This is a straightforward separable variables question requiring standard separation, integration of simple functions (1/y and (1-2x²)/x), and application of initial conditions. The algebra is routine and the 'no logarithms' requirement simply means exponentiating, which is a standard final step. Slightly easier than average due to its mechanical nature with no conceptual challenges.
4 The variables \(x\) and \(y\) satisfy the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = y \left( 1 - 2 x ^ { 2 } \right)$$
and it is given that \(y = 2\) when \(x = 1\). Solve the differential equation and obtain an expression for \(y\) in terms of \(x\) in a form not involving logarithms.
Separate variables and attempt integration of at least one side
M1*
Obtain term \(\ln y\)
A1
Obtain terms \(\ln x - x^2\)
A1
Use \(x = 1\) and \(y = 2\) to evaluate a constant, or as limits
DM1*
Obtain correct solution in any form, e.g. \(\ln y = \ln x - x^2 + \ln 2 + 1\)
A1
Obtain correct expression for \(y\), free of logarithms: \(y = 2x\exp(1-x^2)\)
A1
Total
[6]
## Question 4:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Separate variables and attempt integration of at least one side | M1* | |
| Obtain term $\ln y$ | A1 | |
| Obtain terms $\ln x - x^2$ | A1 | |
| Use $x = 1$ and $y = 2$ to evaluate a constant, or as limits | DM1* | |
| Obtain correct solution in any form, e.g. $\ln y = \ln x - x^2 + \ln 2 + 1$ | A1 | |
| Obtain correct expression for $y$, free of logarithms: $y = 2x\exp(1-x^2)$ | A1 | |
| **Total** | **[6]** | |
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4 The variables $x$ and $y$ satisfy the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = y \left( 1 - 2 x ^ { 2 } \right)$$
and it is given that $y = 2$ when $x = 1$. Solve the differential equation and obtain an expression for $y$ in terms of $x$ in a form not involving logarithms.
\hfill \mbox{\textit{CAIE P3 2016 Q4 [6]}}