Discussion
A solution of the differential equation
\[ xy' = 2y \]
passing through (1, 2), also passes through :
Answer: C
The solution also passes through (2, 8).
Separate variables: \(\dfrac{dy}{y} = \dfrac{2\, dx}{x}\), so \(\ln|y| = 2\ln|x| + c\) and \(y = Cx^{2}\). The point (1, 2) gives \(C = 2\), so \(y = 2x^{2}\).
At \(x = 2\), \(y = 2 \times 4 = 8\).
(2, 4) fits \(y = x^{2}\), the curve through (1, 1), not (1, 2). (2, 16) would need \(y = 2x^{3}\), and (2, 1) lies on none of the solution curves through (1, 2).
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