Find the Solution to the Differential Equation $\frac { d y } { d x } = \frac { \cos ^ { 2 } y } { x }$

question 15

Multiple Choice

Find the solution to the differential equation $\frac { d y } { d x } = \frac { \cos ^ { 2 } y } { x }$ with $y ( 1 ) = - \frac { \pi } { 4 }$ .

• A) \(y = \tan ^ { - 1 } ( \ln | x | + 1 )\)
• B) \(y = \tan ^ { - 1 } ( \ln | x | - 1 )\)
• C) \(y = \sin ^ { - 1 } \left( \ln | x | + \frac { \sqrt { 2 } } { 2 } \right)\)
• D) \(y = \cos ^ { - 1 } \left( \ln | x | + \frac { \sqrt { 2 } } { 2 } \right)\)

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Definitions:

\(3 ^ { n }\)

An exponential expression where 3 is the base and \(n\) is the exponent, indicating three raised to the power of \(n\).

\(n + 4\)

An algebraic expression indicating the sum of the variable 'n' and the number four.

\(- \frac { 3 } { n + 4 }\)

A fraction representing negative three divided by the sum of n and four, where n is any real number.

\(a _ { 5 }\)

The fifth term in a sequence, often defined by a formula or recurrence relation.