Let y=y(x) be the solution curve of the differential equation sec y (d y)/(d x)+2 x sin y=x³ cos y, y(1)=0. Then y(√3) is equal to:

Solution

Maths sir · 5 steps

Divide by cos y and put t = tan y; this mixed equation turns into a plain linear one.

1. Divide by cos y. Dividing every term by cos y gives sec² y dy/dx + 2x tan y = x³. Since d/dx(tan y) = sec² y dy/dx, put t = tan y to get dt/dx + 2xt = x³.
2. Integrating factor. Here P = 2x, so the integrating factor is e^(∫ 2x dx) = e^(x²). That gives t e^(x²) = ∫ x³ e^(x²) dx.
3. Do the integral. Put u = x², so x dx = du/2. The integral becomes 1/2∫ u eᵘ du = 1/2(u-1)eᵘ + C.

   tan y = (x² - 1)/2 + C e^(-x²)

4. Use y(1) = 0. At x = 1, tan 0 = 0 = 0 + Ce⁻¹, so C = 0. The solution is tan y = (x²-1)/2.
5. So. At x = √3, tan y = (3-1)/2 = 1, so y(√3) = π/4, option (d).

The idea to keep. If a trig function of y sits with dy/dx and its derivative appears too, divide to make that derivative show up, then substitute. Most such equations become linear in the new variable.

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