CAIE A-Level Mathematics 3.8.3 Initial Conditions and Particular Solutions

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.

Exam points

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

[Maximum number: 7]

The variables x and y satisfy the differential equation
$\frac{dy}{dx}=\frac{1+4y^2}{e^x}.$
It is given that y=0 when x=1.

Solve the differential equation, obtaining an expression for y in terms of x.

Separate variables and attempt integration:
$\int\frac{1}{1+4y^2}\,dy=\int e^{-x}\,dx.$
B1

Obtain a term of the form $a\tan^{-1}(2y)$ .
M1

$\frac12\tan^{-1}(2y)$
A1

Obtain
$-e^{-x}.$
B1

Use x=1, y=0 to evaluate the constant.
M1

Obtain a correct answer in any form.
A1

Final answer:
$y=\frac12\tan\left(2e^{-1}-2e^{-x}\right).$
A1

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3.8.3—An initial condition to find a question 1