Question

If \[{\rm{n}}\] is any integer, then the general solution of the equation\[\cos x - \sin x = \frac{1}{{\sqrt 2 }}\]is
A. \[x = 2n\pi - \dfrac{\pi }{{12}}\]or \[x = 2n\pi + \dfrac{{7\pi }}{{12}}\]
В. \[x = n\pi \pm \dfrac{\pi }{{12}}\]
C. \[x = 2n\pi + \dfrac{\pi }{{12}}\]or\[x = 2n\pi - \dfrac{{7\pi }}{{12}}\]
D. \[x = n\pi + \dfrac{\pi }{{12}}\]or\[x = n\pi - \dfrac{{7\pi }}{{12}}\]

Answer 1

Hint: A differential equation is created as follows if an equation in independent and dependent variables involving some arbitrary constants is given: As many times as, there are arbitrary constants in the preceding equation, differentiate it with respect to the independent variable (let's say x). Get rid of the random constants. The necessary differential equation is the eliminator. In this situation, the equation\[\cos x - \sin x = \frac{1}{{\sqrt 2 }}\]can be resolved by first dividing it by the denominator, then rewriting it with respect to cos, and finally solving as a result. We should simplify the equation using algebraic identity for the product of sum and difference of two numbers to form two equations.





Complete step by step solution:We have given the equation, as per the question as:
\[\cos x - \sin x = \dfrac{1}{{\sqrt 2 }}\]
Dividing the whole equation by\[\sqrt 2 \]:
\[\dfrac{1}{{\sqrt 2 }}\cos x - \dfrac{1}{{\sqrt 2 }}\sin x = \dfrac{1}{2}\]
Rewrite the obtained equation as:
\[ \Rightarrow \cos \left( {\dfrac{\pi }{4} + x} \right) = \cos \dfrac{\pi }{3}\]
Solve with respect to cosine function:
\[ \Rightarrow \dfrac{\pi }{4} + x = 2n\pi \pm \dfrac{\pi }{3}\]
On simplifying the resultant equation, we obtain,
\[x = 2n\pi + \dfrac{\pi }{3} - \dfrac{\pi }{4} = 2n\pi + \dfrac{\pi }{{12}}\]
Or
\[x = 2n\pi - \dfrac{\pi }{3} - \dfrac{\pi }{4} = 2n\pi - \dfrac{{7\pi }}{{12}}\]
Hence, the general solution of the equation
\[\cos x - \sin x = \dfrac{1}{{\sqrt 2 }}\]is \[x = 2n\pi + \dfrac{\pi }{{12}}\]or \[x = 2n\pi - \dfrac{{7\pi }}{{12}}\].



Option ‘C’ is correct

Note: Sometimes, when tackling differential equation issues, students fail to eliminate the arbitrary constants that serve as the variables' coefficients, which produce a whole different conclusion. So, when creating differential equations, be cautious to exclude any arbitrary constants or parameters. Keep in mind that the specific solution is one in which the integral constant is given a numerical value.