Date | None Specimen | Marks available | 7 | Reference code | SPNone.3ca.hl.TZ0.2 |
Level | HL only | Paper | Paper 3 Calculus | Time zone | TZ0 |
Command term | Find | Question number | 2 | Adapted from | N/A |
Question
Consider the differential equation
xdydx=y+√x2−y2, x>0, x2>y2.
Show that this is a homogeneous differential equation.
[1]
a.
Find the general solution, giving your answer in the form y=f(x) .
[7]
b.
Markscheme
the equation can be rewritten as
dydx=y+√x2−y2x=yx+√1−(yx)2 A1
so the differential equation is homogeneous AG
[1 mark]
a.
put y = vx so that dydx=v+xdvdx M1A1
substituting,
v+xdvdx=v+√1−v2 M1
∫dv√1−v2=∫dxx M1
arcsinv=lnx+C A1
yx=sin(lnx+C) A1
y=xsin(lnx+C) A1
[7 marks]
b.
Examiners report
[N/A]
a.
[N/A]
b.