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A linear differential equation of the first order is of the form dy/dx + Py = Q or dx/dy + Px = Q, where P and Q are constants or functions of x (or y respectively). The dependent variable and its derivative must appear only in the first degree and not be multiplied together.
Rearranging (1+x^2)dy + 2xy dx = cot x dx gives dy/dx + (2x/(1+x^2))y = cot x / (1+x^2). This is linear in y.
Expanding y + d/dx(xy) = x(sin x + log x) gives y + y + x(dy/dx) = x(sin x + log x), which simplifies to x(dy/dx) + 2y = x(sin x + log x). This is linear in y.
Rearranging x(1+y^2)dx - y(1+x^2)dy = 0 gives dx/dy = y(1+x^2) / (x(1+y^2)). This is a variable separable equation, but it is not linear because the term x^2 appears in the numerator, preventing the form dx/dy + Px = Q.
Rearranging y dx - (x + 3y^2) dy = 0 gives dx/dy = (x + 3y^2) / y, which simplifies to dx/dy - (1/y)x = 3y. This is linear in x.
Final Answer: C