NCERT Class 12 · Ch 9 Exercise 9.4 · Q1
NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 1
Show the equation y′=\(\frac{x^{2} + y^{2}}{x^{2} + x y}\) 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: −y/x−2log|1−y/x|−log|x|=C, plus y=x.
- 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=\(- \frac{v - 1}{v + 1}\). Separate dv/(\(- \frac{v - 1}{v + 1}\))=dx/x on branches where division is valid.
- Integration and back-substitution give \(- \log{\left(\left|{x}\right| \right)} - 2 \log{\left(\left|{1 - \frac{y}{x}}\right| \right)} - \frac{y}{x}\) = C.
- The divided case v=1 gives the additional straight-line solution y=x; the original slope also excludes x=0 and y=−x.
- Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.
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