First order differential equations (integrating factor)
7
Show that an appropriate integrating factor for
$$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$
is \(x + \sqrt { x ^ { 2 } - 1 }\).
Hence find the solution of the differential equation
$$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$
for which \(y = 1\) when \(x = \frac { 5 } { 4 }\). Give your answer in the form \(y = f ( x )\).