CAIE A-Level Mathematics 3.8.3 Initial Conditions and Particular Solutions

CAIE A-Level Mathematics 3.8.3 Initial Conditions and Particular Solutions
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise using one or more initial conditions to determine integration and model constants, obtain a particular differential-equation solution and evaluate a target value or time.

How this is tested

  • substitute the given x, y or t values into the integrated general solution to find c
  • use a second condition where necessary to determine a proportionality or model constant
  • evaluate the completed particular solution at the target time, depth or coordinate

Question 8

[Maximum number: 8]

The variables x and y satisfy the differential equation

(x2+1)dy dx=kxe2y,\left(x^{2}+1\right) \frac{\mathrm{d} y}{\mathrm{~d} x}=k x \mathrm{e}^{2 y},

where k is a constant. It is given that y=0 when x=0 and that y=−12y=-\frac{1}{2} when x=1.
Solve the differential equation and find the exact value of y when x=3x=\sqrt{3}.