(i) Separate variables and integrate: \(2y^{\frac{1}{2}} = \ldots\)

Integrate by parts for \(x \sin \left( \frac{1}{3}x \right)\):

\(-3x \cos \left( \frac{1}{3}x \right) + 9 \sin \left( \frac{1}{3}x \right)\)

General solution: \(y = \left( -\frac{3}{10}x \cos \left( \frac{1}{3}x \right) + \frac{9}{10} \sin \left( \frac{1}{3}x \right) + c \right)^2\)

(ii) Use \(x = 0\) and \(y = 100\) to find constant:

Substitute \(x = 25\) to find \(y\):

\(y = 203\)