Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations
Class 12 Maths · Differential Equations · Long Answer #51
Solve y′ + sec²x · y = tan x sec²x, y(0) = 0.
✅ Step-by-Step Solution
Final Answer: y = tan x − 1 + e^(−tan x), −π/2 < x < π/2.
- IF = exp(∫sec²x dx) = e^(tan x).
- Then ye^(tan x) = ∫tan x sec²x e^(tan x)dx + C. Put t = tan x; ∫teᵗdt = (t − 1)eᵗ.
- Thus y = tan x − 1 + Ce^(−tan x). At x = 0, C = 1.
- Therefore y = tan x − 1 + e^(−tan x) on (−π/2, π/2).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] IF = exp(∫sec²x dx) = e^(tan x).
- [1 mark] Then ye^(tan x) = ∫tan x sec²x e^(tan x)dx + C. Put t = tan x; ∫teᵗdt = (t − 1)eᵗ.
- [1 mark] Thus y = tan x − 1 + Ce^(−tan x). At x = 0, C = 1.
- [0.5 marks] Therefore y = tan x − 1 + e^(−tan x) on (−π/2, π/2).
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