NCERT Class 12 · Ch 9 Exercise 9.4 · Q4
NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 4
Show the equation y′=\(\frac{- x^{2} + y^{2}}{2 x y}\) is homogeneous and solve it.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: x²+y²=Cx, on finite-slope portions of the original curve.
- After division the slope depends only on y/x (on a fixed sign component when radicals occur). Put y=vx, so y′=v+xv′.
- The reduced equation is x dv/dx=\(- \frac{v^{2} + 1}{2 v}\). Separate dv/(\(- \frac{v^{2} + 1}{2 v}\))=dx/x on branches where division is valid.
- Integration and back-substitution give \(\frac{x^{2} + y^{2}}{x}\) = C.
- Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.
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