Class 12 Differential Equations · Very Long Answer5 Marksseparable_differential_equations
Class 12 Maths · Differential Equations · Very Long Answer #78
Solve (1 + e^(2x))dy + (1 + y²)eˣdx = 0, y(0) = 1.
✅ Step-by-Step Solution
Final Answer: y = e^(−x), x ∈ ℝ.
- Separate dy/(1 + y²) = −eˣdx/(1 + e^(2x)).
- Put t = eˣ, dt = eˣdx. Integrate to arctan y = −arctan(eˣ) + C.
- At (0, 1), π/4 = −π/4 + C gives C = π/2.
- Thus arctan y + arctan(eˣ) = π/2. Since eˣ > 0, y is its positive reciprocal.
- Therefore y = e^(−x), which directly satisfies the original equation and y(0) = 1.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Separate dy/(1 + y²) = −eˣdx/(1 + e^(2x)).
- [1 mark] Put t = eˣ, dt = eˣdx. Integrate to arctan y = −arctan(eˣ) + C.
- [1 mark] At (0, 1), π/4 = −π/4 + C gives C = π/2.
- [1 mark] Thus arctan y + arctan(eˣ) = π/2. Since eˣ > 0, y is its positive reciprocal.
- [1 mark] Therefore y = e^(−x), which directly satisfies the original equation and y(0) = 1.
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