Answer: dy/dx - y/x = x (linear in y).
- A dy/dx - y/x = x (linear in y)
- B y dy/dx = x, an unrelated separable rearrangement
- C dy/dx = x², ignoring the y term on the left side
- D dy/dx + y/x = x, with the sign on y/x flipped
Correct answer: A. dy/dx - y/x = x (linear in y)
Explanation: Divide by x: dy/dx − y/x = x. Linear first-order with P(x)=−1/x, Q(x)=x. Integrating factor: e<sup>∫−1/x dx</sup>=e<sup>−lnx</sup>=1/x. Multiply: d(y/x)/dx=1. Integrate: y/x=x+C, so y=x²+Cx.
The general solution y=Ax² represents an entire FAMILY of curves, one for each value of the arbitrary constant A; a particular solution (fixed by an initial condition) selects exactly one curve from this family.
Concept context
Equations involving derivatives and their solutions