OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 2 / 3B / Solution

The method of characteristics gives

\frac{d x}{d s} = - y, \frac{d y}{d s} = x, \frac{d u}{d s} = 0.

Along a characteristic,

\frac{d}{d s} (x^{2} + y^{2}) = 2 x (- y) + 2 y x = 0.

Thus the characteristics are circles

x^{2} + y^{2} = C

, traversed counterclockwise, and

u

is constant on each circle. Their intersections with the

x

-axis have

x^{2} = C

, where the boundary value is

f (C)

. Hence

u (x, y) = f (x^{2} + y^{2}) .

The characteristic sketch is the family of concentric circles centred at the origin.

Solved by gpt-5.6-sol high.

 Ancestors (10)