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Rates of change and differential equations — Question 15

QCAA 2024, Paper 1 · 6 marks

Q15 · 2024 · Technology-freeSimple familiar6 marks

QUESTION 15 (6 marks)

A sketch of a partially completed slope field for the differential equation dydx=1−y\frac{dy}{dx}=1-y is shown.
Partially completed slope field for dy/dx=1-y
a)
Complete the slope field by sketching the slopes at the 10 points indicated.
[2 marks]
b)
Use the completed slope field to sketch the solution curve for dydx=1−y\frac{dy}{dx}=1-y, given the point (−1,0)(-1,0) lies on this curve.
[1 mark]
c)
The differential equation can be solved by rearranging it into the form 11−ydydx=1\frac{1}{1-y}\frac{dy}{dx}=1. Determine the equation of the solution curve sketched in Question 15b) by solving the differential equation, given the point (−1,0)(-1,0) lies on this curve. Leave your answer in the form y=f(x)y=f(x).
[3 marks]
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