CAIE A-Level Mathematics A2 3.8.3 An Initial Condition to Find a Questions

Practise solving differential equations with initial conditions to obtain particular solutions.

Exam points

apply initial conditions after solving differential equations
determine constants in particular solutions
evaluate particular solutions at specified values

[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:
$\int\frac{1}{1+4y^2}\,dy=\int e^{-x}\,dx.$
Obtain a term of the form $a\tan^{-1}(2y)$ .
$\frac12\tan^{-1}(2y)$
Obtain
$-e^{-x}.$
Use x=1, y=0 to evaluate the constant.
Obtain a correct answer in any form.
Final answer:
$y=\frac12\tan\left(2e^{-1}-2e^{-x}\right).$

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CAIE A-Level Mathematics A2 3.8.3 An Initial Condition to Find a Questions question 1