Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #45
Solve xy′ + x cos²(y/x) = y, y(1) = π/4.
✅ Step-by-Step Solution
Final Answer: y = x arctan(1 − ln x), x > 0.
- Put v = y/x; y′ = v + xv′. Then xv′ = −cos²v.
- Near v(1) = π/4, integrate sec²v dv = −dx/x to obtain tan v = −ln x + C.
- The condition tan(π/4) = 1 gives C = 1. The continuous branch through π/4 is v = arctan(1 − ln x).
- Thus y = x arctan(1 − ln x), x > 0.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Put v = y/x; y′ = v + xv′. Then xv′ = −cos²v.
- [1 mark] Near v(1) = π/4, integrate sec²v dv = −dx/x to obtain tan v = −ln x + C.
- [1 mark] The condition tan(π/4) = 1 gives C = 1. The continuous branch through π/4 is v = arctan(1 − ln x).
- [0.5 marks] Thus y = x arctan(1 − ln x), x > 0.
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