First order differential equations (integrating factor)
3
Show that \(x ^ { 2 }\) is an integrating factor for the first-order differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
Solve this differential equation, given that \(y = 1\) when \(x = 2\).