Let x=x(y) be the solution of the differential equation y=(x-y (d x)/(d y)) sin (x/y), y>0 and x(1)=π/2. Then cos (x(2)) is equal to :

Solution

Maths sir · 5 steps

Since x/y sits inside the sine, put x = vy, and the equation separates into sin v dv = -dy/y.

1. Substitute x = vy. Then dx/dy = v + ydv/dy. The bracket becomes x - ydx/dy = vy - vy - y²dv/dy = -y²dv/dy.
2. Separate variables. The equation is now y = -y²dv/dy sin v. Divide by y² (allowed because y > 0) and separate.

   ∫ sin v dv = -∫ dy/y ⇒ -cos v = -ln y + C

3. Fix the constant. At y = 1, v = π/2, so cos v = 0 and ln 1 = 0. That gives C = 0, so the solution is cos(x/y) = ln y.
4. Put y = 2. cos(x(2)/2) = ln 2. The question asks for cos x, not cos x/2, so use the double-angle formula cos x = 2 cos² x/2 - 1.
5. So. cos(x(2)) = 2(ln 2)² - 1, which is option (d).

The idea to keep. Whenever x/y appears inside a function, try x = vy. Then check whether the question wants that same angle or a multiple of it before you stop.

Answer from the NTA final answer key. Solution by PrepWiser.