First order differential equations (integrating factor)
7
Show that an appropriate integrating factor for
$$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$
is \(x + \sqrt { x ^ { 2 } + 1 }\).
Hence find the solution of the differential equation
$$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$
for which \(y = \ln 2\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).