 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-2/3/solution

The equation is not separable, because

x^{2} - x y = x (x - y)

cannot be written as a product of a function of

x

and a function of

y

on an open region. For example, the ratio

(x^{2} - x y_{1}) / (x^{2} - x y_{2}) = (x - y_{1}) / (x - y_{2})

depends on

x

for distinct fixed

y_{1}, y_{2}

, whereas a separated product would give a constant ratio.

It is not homogeneous in the sense of a homogeneous first-order differential equation: the right-hand side is not a function of

y / x

x y

term to the left leaves a nonzero forcing.

It is linear, since it has the standard linear ordinary differential equation form

y^{'} + x y = x^{2} .

Thus the classifications are (a) no, (b) no, (c) yes.

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