Let y=y(x) be the solution of the differential equation
(d y)/(d x)+3(tan ² x) y+3 y=sec ² x, y(0)=1/3+e³. Then y(π/4) is equal to :

Solution

Maths sir · 4 steps

Join the two y-terms: 3 tan² x + 3 = 3 sec² x, and the equation becomes a clean linear one.

1. Combine the y-terms. 3(tan² x)y + 3y = 3(1 + tan² x)y = 3(sec² x)y. So the equation is dy/dx + 3 sec² x y = sec² x, which is linear with P = 3 sec² x and Q = sec² x.
2. Integrating factor. ∫ 3 sec² x dx = 3 tan x, so the integrating factor is e³ᵗᵃⁿˣ. To integrate the right side, put t = 3 tan x, so dt = 3 sec² x dx.

   y e³ᵗᵃⁿˣ = ∫ sec² x e³ᵗᵃⁿˣ dx = 1/3e³ᵗᵃⁿˣ + C

3. Use y(0). Divide by e³ᵗᵃⁿˣ to get y = 1/3 + C e⁻³ᵗᵃⁿˣ. At x = 0 this gives y = 1/3 + C. This must equal 1/3 + e³, so C = e³.
4. So. y(π/4) = 1/3 + e³ · e^(-3 tan(π/4)) = 1/3 + e³ · e⁻³ = 1/3 + 1 = 4/3. The answer is (a).

The idea to keep. When a linear equation has messy trig coefficients, first look for an identity that merges them. Here 1 + tan² x = sec² x turns the problem into a standard one with integrating factor e³ᵗᵃⁿˣ.

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