 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2022/3/solution

P d x + Q d y

, equality of mixed partial derivatives requires

P_{y} = Q_{x}

. The matching pair is

P = f_{1}

Q = f_{3}

, since

\partial_{y} f_{1} = \frac{c o s y}{x} - e^{x} sin 2 y = \partial_{x} f_{3} .

The other pairings do not give the same partial derivatives on an open region. Integrating

f_{1}

with respect to

x

gives

ln x sin y + e^{x} cos^{2} y + C (y)

. Its

y

derivative agrees with

f_{3}

precisely when

C^{'} (y) = 0

. Hence

f (x, y) = ln x sin y + e^{x} cos^{2} y + C, x > 0.

This also proves existence, rather than relying only on the necessary mixed-partial derivative test.

 New to topics? Read the docs here!