NCERT Class 12 · Ch 9 Exercise 9.4 · Q6
NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 6
Show xdy−ydx=√(x²+y²)dx is homogeneous and solve.
💡 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=(C x²−1/C)/2, C>0, on x≠0.
- With y=vx, x²v′=|x|√(1+v²), so dv/√(1+v²)=dx/|x|.
- On x>0 integrate asinh v=log x+K. On x<0 integrate asinh v=−log|x|+K.
- Both families can be written y=(Cx²−1/C)/2 with C>0. Direct substitution gives xy′−y=(Cx²+1/C)/2=√(x²+y²).
- The sign C>0 is essential because the radical is nonnegative.
- Each displayed parabola also extends smoothly through x=0 at y=−1/(2C)<0 and satisfies the original undivided equation there. The homogeneous division method itself uses x≠0.
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