Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations
Class 12 Maths · Differential Equations · Long Answer #62
Show that xy′sin(y/x) + x − y sin(y/x) = 0 is homogeneous and solve it with y(1) = π/2.
✅ Step-by-Step Solution
Final Answer: cos(y/x) = ln x; the branch through the initial point is y = x arccos(ln x), e^(−1) < x < e.
- Rearrange to y′ = y/x − csc(y/x); the right-hand side depends only on y/x, so it is homogeneous.
- Put v = y/x. Then xv′ = −csc v, so −sin v dv = dx/x.
- Integrating gives cos v = ln|x| + C. The initial point gives C = 0.
- Thus y = x arccos(ln x) on (e^(−1), e), the branch through π/2; endpoints have sin(y/x) = 0 and are excluded.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Rearrange to y′ = y/x − csc(y/x); the right-hand side depends only on y/x, so it is homogeneous.
- [1 mark] Put v = y/x. Then xv′ = −csc v, so −sin v dv = dx/x.
- [1 mark] Integrating gives cos v = ln|x| + C. The initial point gives C = 0.
- [0.5 marks] Thus y = x arccos(ln x) on (e^(−1), e), the branch through π/2; endpoints have sin(y/x) = 0 and are excluded.
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