If sin (y/x)=log ₑ|x|+α/2 is the solution of the differential equation x cos (y/x) (d y)/(d x)=y cos (y/x)+x and y(1)=π/3, then α² is equal to

Solution

Maths sir · 4 steps

Everything sits as y/x, so put y = vx and the equation separates into cos v dv = dx/x.

1. Substitute y = vx. Every term depends on y/x, so the equation is homogeneous. Put y = vx, which gives dy/dx = v + xdv/dx.
2. Cancel and separate. The equation becomes x cos v(v + xdv/dx) = vx cos v + x. The vx cos v terms cancel, leaving x²cos v dv/dx = x.

   cos v dv = dx/x ⇒ sin(y/x) = logₑ|x| + C

3. Use y(1). At x = 1, y = π/3, so sin π/3 = 0 + α/2. That gives √3/2 = α/2, so α = √3.
4. So. α² = 3, which is option (d).

The idea to keep. When dy/dx is a function of y/x alone, put y = vx. The v terms usually cancel and the equation separates. Use the initial condition only at the end.

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