Class 12 Differential Equations · Very Long Answer5 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #107
Solve sin 2x · y′ − y = tan x, y(π/4) = 0.
✅ Step-by-Step Solution
Final Answer: y = tan x − √(tan x), 0 < x < π/2.
- Divide by sin 2x: P = −1/sin 2x. On (0, π/2), IF = exp(−½ln(tan x)) = 1/√(tan x).
- Then [y/√(tan x)]′ = tan x/[sin 2x √(tan x)].
- Put t = √(tan x). Since t′ = sec²x/(2√(tan x)), the right side is t′. Hence y/√(tan x) = √(tan x) + C.
- At x = π/4, y = 0 gives C = −1.
- Thus y = tan x − √(tan x), on (0, π/2).
📋 CBSE Stepwise Marking Scheme
- [1 mark] Divide by sin 2x: P = −1/sin 2x. On (0, π/2), IF = exp(−½ln(tan x)) = 1/√(tan x).
- [1 mark] Then [y/√(tan x)]′ = tan x/[sin 2x √(tan x)].
- [1 mark] Put t = √(tan x). Since t′ = sec²x/(2√(tan x)), the right side is t′. Hence y/√(tan x) = √(tan x) + C.
- [1 mark] At x = π/4, y = 0 gives C = −1.
- [1 mark] Thus y = tan x − √(tan x), on (0, π/2).
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