 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/13/b/solution

Let

M = x + y

and

N = x

. Their cross partial derivatives agree:

M_{y} = N_{x} = 1

. More directly, construct the scalar function

F (x, y) = \frac{x ^{2}}{2} + x y, d F = (x + y) d x + x d y .

This exhibits the expression as an exact differential. Along a solution of the differential equation,

\frac{d}{d x} F (x, y (x)) = x + y + x y^{'} = 0.

\frac{x ^{2}}{2} + x y = C, y = \frac{C}{x} - \frac{x}{2} (x \neq = 0) .

For

C = 0

y = - x /2

also extends smoothly through

x = 0

and satisfies the original, undivided equation there. Nonzero

C

cannot extend through zero as a finite-valued differentiable solution. The extra vertical component

x = 0

of the implicit zero-level equation is not a graph

y (x)

on an interval and is not an additional solution of this ordinary differential equation.

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