First order differential equations (integrating factor)
10
By writing \(y = \sec ^ { 3 } \theta\) as \(\frac { 1 } { \cos ^ { 3 } \theta }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = 3 \sin \theta \sec ^ { 4 } \theta\).
The variables \(x\) and \(\theta\) satisfy the differential equation
$$\left( x ^ { 2 } + 9 \right) \sin \theta \frac { d \theta } { d x } = ( x + 3 ) \cos ^ { 4 } \theta$$
It is given that \(x = 3\) when \(\theta = \frac { 1 } { 3 } \pi\).
Solve the differential equation to find the value of \(\cos \theta\) when \(x = 0\). Give your answer correct to 3 significant figures.
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