NCERT Class 12 · Ch 9 Miscellaneous Exercise · Q3
NCERT Class 12 Maths · Chapter 9 Differential Equations · Miscellaneous Exercise Question 3
Prove x²−y²=c(x²+y²)² solves (x³−3xy²)dx=(y³−3x²y)dy.
💡 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: \(\frac{x^{2} - y^{2}}{\left(x^{2} + y^{2}\right)^{2}}\) = C.
- Solve for the parameter: c=(x²−y²)/(x²+y²)², away from the origin.
- Differentiating this invariant gives −2(x³−3xy²)dx+2(y³−3x²y)dy=0 after multiplication by (x²+y²)³.
- This is precisely the stated equation. One independent parameter gives the local general family on regular finite-slope branches.
- Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.
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