Standard +0.8 This is a standard integrating factor question from Further Maths FP3, requiring identification of the integrating factor x^3, multiplication through, integration of a non-trivial expression (x^4+3)^(3/2), and application of initial conditions. While the method is routine for Further Maths students, the integration step requires either substitution or recognition, and the algebraic manipulation across 9 marks elevates it slightly above average difficulty.
4 By using an integrating factor, find the solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 3 } { x } y = \left( x ^ { 4 } + 3 \right) ^ { \frac { 3 } { 2 } }$$
given that \(y = \frac { 1 } { 5 }\) when \(x = 1\).
(9 marks)