Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #54
Solve y′ + 2y tan x = sin x, y(π/3) = 0.
✅ Step-by-Step Solution
Final Answer: y = cos x − 2cos²x, −π/2 < x < π/2.
- An integrating factor is exp(∫2tan x dx) = sec²x.
- Then (y sec²x)′ = sin x sec²x = (sec x)′. Integrating yields y sec²x = sec x + C.
- At x = π/3, y = 0 and sec x = 2, so C = −2.
- Thus y = cos x − 2cos²x on the interval (−π/2, π/2).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] An integrating factor is exp(∫2tan x dx) = sec²x.
- [1 mark] Then (y sec²x)′ = sin x sec²x = (sec x)′. Integrating yields y sec²x = sec x + C.
- [1 mark] At x = π/3, y = 0 and sec x = 2, so C = −2.
- [0.5 marks] Thus y = cos x − 2cos²x on the interval (−π/2, π/2).
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