First order differential equations (integrating factor)
7
Show that
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \frac { x } { 2 } \sqrt { x ^ { 2 } - 9 } - \frac { 9 } { 2 } \cosh ^ { - 1 } \frac { x } { 3 } \right) = \sqrt { x ^ { 2 } - 9 }$$
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Find the solution of the differential equation
$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = x ^ { 2 } \sqrt { x ^ { 2 } - 9 }$$
given that \(y = 1\) when \(x = 3\). Give your answer in the form \(y = \mathrm { f } ( x )\).
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