Standard +0.3 This is a straightforward separable differential equation requiring standard technique: separate variables, integrate both sides (including a simple substitution for the right side), apply initial condition to find the constant, and rearrange. While it requires multiple steps for 7 marks, it follows a completely standard procedure with no conceptual challenges or novel insights needed—slightly easier than the average A-level question.
The coordinates \((x, y)\) of a general point on a curve satisfy the differential equation
$$x\frac{dy}{dx} = (2 - x^2)y.$$
The curve passes through the point \((1, 1)\). Find the equation of the curve, obtaining an expression for \(y\) in terms of \(x\). [7]
Separate variables correctly and integrate at least one side
B1
Obtain term ln y
B1
1
Obtain terms 2 ln x – x2
Answer
Marks
2
B1+B1
Use x = 1, y = 1 to evaluate a constant, or as limits
M1
Obtain correct solution in any form, e.g.
1 1
lny=2lnx− x2 +
Answer
Marks
2 2
A1
1 1
Rearrange as y=x2exp − x2 , or equivalent
Answer
Marks
2 2
A1
7
Answer
Marks
Guidance
Question
Answer
Marks
Question 5:
5 | Separate variables correctly and integrate at least one side | B1
Obtain term ln y | B1
1
Obtain terms 2 ln x – x2
2 | B1+B1
Use x = 1, y = 1 to evaluate a constant, or as limits | M1
Obtain correct solution in any form, e.g.
1 1
lny=2lnx− x2 +
2 2 | A1
1 1
Rearrange as y=x2exp − x2 , or equivalent
2 2 | A1
7
Question | Answer | Marks | Guidance
The coordinates $(x, y)$ of a general point on a curve satisfy the differential equation
$$x\frac{dy}{dx} = (2 - x^2)y.$$
The curve passes through the point $(1, 1)$. Find the equation of the curve, obtaining an expression for $y$ in terms of $x$. [7]
\hfill \mbox{\textit{CAIE P3 2018 Q5 [7]}}