Moderate -0.3 This is a straightforward separable variables question requiring standard technique: separate variables, integrate both sides (including ln|y| and ln(x) - x²), apply initial conditions. The integration is routine and the algebra is manageable. Slightly easier than average due to being a textbook-standard separable DE with no complications.
8 The coordinates \(( x , y )\) of a general point of a curve satisfy the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 1 - 2 x ^ { 2 } \right) y$$
for \(x > 0\). It is given that \(y = 1\) when \(x = 1\).
Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\).
Separate variables correctly and attempt integration of at least one side
B1
\(\dfrac{1}{y}\,dy=\dfrac{1-2x^2}{x}\,dx\)
Obtain term \(\ln y\)
B1
Obtain terms \(\ln x - x^2\)
B1
Use \(x=1,\ y=1\) to evaluate a constant, or as limits, in a solution containing at least 2 terms of the form \(a\ln y\), \(b\ln x\) and \(cx^2\)
M1
The 2 terms of required form must be from correct working e.g. \(\ln y = \ln x - x^2 + 1\)
Obtain correct solution in any form
A1
Rearrange and obtain \(y=xe^{1-x^2}\)
A1
OE
6
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Separate variables correctly and attempt integration of at least one side | B1 | $\dfrac{1}{y}\,dy=\dfrac{1-2x^2}{x}\,dx$ |
| Obtain term $\ln y$ | B1 | |
| Obtain terms $\ln x - x^2$ | B1 | |
| Use $x=1,\ y=1$ to evaluate a constant, or as limits, in a solution containing at least 2 terms of the form $a\ln y$, $b\ln x$ and $cx^2$ | M1 | The 2 terms of required form must be from correct working e.g. $\ln y = \ln x - x^2 + 1$ |
| Obtain correct solution in any form | A1 | |
| Rearrange and obtain $y=xe^{1-x^2}$ | A1 | OE |
| | **6** | |
8 The coordinates $( x , y )$ of a general point of a curve satisfy the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 1 - 2 x ^ { 2 } \right) y$$
for $x > 0$. It is given that $y = 1$ when $x = 1$.\\
Solve the differential equation, obtaining an expression for $y$ in terms of $x$.\\
\hfill \mbox{\textit{CAIE P3 2020 Q8 [6]}}