Let y=y(x) be the solution of the differential equation (d y)/(d x)+2 y sec ² x=2 sec ² x+3 tan x · sec ² x such that y(0)=5/4. Then 12(y(π/4)-e⁻²) is equal to_____________________

Solution

Maths sir · 5 steps

Every term carries sec² x, so put t = tan x and this becomes an easy linear equation in t.

1. Integrating factor. This is linear in y with P = 2 sec² x. So the integrating factor is e^(∫ 2 sec² x dx) = e²ᵗᵃⁿˣ.
2. Substitute t. Multiply through by e²ᵗᵃⁿˣ and put t = tan x, so dt = sec² x dx. The right side becomes e²ᵗ(2 + 3t) dt.

   y e²ᵗ = ∫ e²ᵗ(2+3t) dt = e²ᵗ + 3t/2e²ᵗ - 3/4e²ᵗ + C

3. General solution. Divide by e²ᵗ: y = 1/4 + 3/2 tan x + C e⁻²ᵗᵃⁿˣ. At x = 0, 5/4 = 1/4 + C, so C = 1.
4. Value at π/4. Here tan π/4 = 1, so y(π/4) = 1/4 + 3/2 + e⁻² = 7/4 + e⁻².
5. So. 12(y(π/4) - e⁻²) = 12 × 7/4 = 21.

The idea to keep. When P(x) and Q(x) share a factor like sec² x, substitute the function whose derivative it is. The integral after the integrating factor then becomes a plain integration by parts.

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