Question

Find a particular solution satisfying the given condition:\(\frac {dy}{dx}-3y\  cot\ x=sin2x;\ y=2 \ when\  x=\frac \pi2\)

Answer 1

The differential equation is \(\frac {dy}{dx}\)3y cot x = sin 2x

This is a linear differential equation of the form:

\(\frac {dy}{dx}\) + py = Q (where p=-3 cot x and Q=sin 2x)

Now, I.F. = ∫(Q×I.F.)dx + C

⇒y.\(\frac {1}{sin^3x}\)=∫[sin2x.\(\frac {1}{sin^3x}\)]dx + C

⇒\(\frac {y}{sin ^3x}\) = 2∫(cot x . cosec x)dx + C

⇒\(\frac {y}{sin ^3x}\) = -2 cosec x + c

⇒y = -2sin²x + C sin³x      ……....(1)

Now, y=2 at x=\(\frac \pi2\)

Therefore,we get:

2=-2+C

⇒C=4

Substituting C=4 in equation(1),we get:

y = -2sin²x + 4sin³x

⇒y = 4sin³x - 2sin²x

This is the required particular solution of the given differential equation.