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Class 12 Differential Equations · Long Answer3 Markslinear_differential_equations

Class 12 Maths · Differential Equations · Long Answer #48

Find the general solution of d(xy²)/dx = 2y(1 + x²).

✅ Step-by-Step Solution

Final Answer: For x ≠ 0: y = 0, or y = 2 + 2x²/5 + C/√|x| on a connected nonzero branch.
  1. Expand to 2xyy′ + y² = 2y(1 + x²). First retain the solution y = 0.
  2. On a nonzero branch, divide by 2xy: y′ + y/(2x) = (1 + x²)/x. Use IF √|x| on x ≠ 0.
  3. Integrating gives √|x|y = √|x|(2 + 2x²/5) + C, since differentiating the right-hand particular product yields √|x|(1 + x²)/x.
  4. Thus y = 2 + 2x²/5 + C/√|x|, together with y = 0. A branch crossing y = 0 is checked in the original undivided equation.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Expand to 2xyy′ + y² = 2y(1 + x²). First retain the solution y = 0.
  • [1 mark] On a nonzero branch, divide by 2xy: y′ + y/(2x) = (1 + x²)/x. Use IF √|x| on x ≠ 0.
  • [1 mark] Integrating gives √|x|y = √|x|(2 + 2x²/5) + C, since differentiating the right-hand particular product yields √|x|(1 + x²)/x.
  • [0.5 marks] Thus y = 2 + 2x²/5 + C/√|x|, together with y = 0. A branch crossing y = 0 is checked in the original undivided equation.

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