First order differential equations

WFM02Topic 2.42 questions

Integrating factors and reducing equations to standard form by substitution.

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Original practice questions written in the style of the exam.

Question 16 marks
Find the solution of the differential equation $\frac{dy}{dx} + y\tan x = \cos^2 x, \qquad -\frac{\pi}{2} < x < \frac{\pi}{2}$ for which $y = 2$ when $x = 0$ . Give your answer in the form $y = f(x)$ .
Mark scheme
- M1 Integrating factor $e^{\int \tan x\,dx} = e^{\ln \sec x}$ (1)

c

x

A1

= \sec x

(1)

M1

\dfrac{d}{dx}(y\sec x) = \cos^2 x \sec x = \cos x

(1)

A1

y\sec x = \sin x + c

(1)

M1 Uses

x = 0

,

y = 2

to find

c = 2

(1)

A1

y = \sin x\cos x + 2\cos x

(1)