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Class 12 Continuity And Differentiability · Long Answer3 Markssecond_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #102

For y=\(\operatorname{atan}^{2}{\left(x \right)}\), prove \(\left(x^{2} + 1\right)^{2}\)y″ + \(2 x \left(x^{2} + 1\right)\)y′ + \(0\)y = \(2\). Assume all open intervals where the displayed expressions are real and differentiable.

✅ Step-by-Step Solution

Final Answer: Identity proved.
  1. Differentiate y=\(\operatorname{atan}^{2}{\left(x \right)}\) to obtain y′=\(\frac{2 \operatorname{atan}{\left(x \right)}}{x^{2} + 1}\).
  2. Differentiate again, using the product and chain rules: y″=\(\frac{2 \left(- 2 x \operatorname{atan}{\left(x \right)} + 1\right)}{\left(x^{2} + 1\right)^{2}}\).
  3. Substitute these derivatives into the left side \(\left(x^{2} + 1\right)^{2}\)y″ + \(2 x \left(x^{2} + 1\right)\)y′ + \(0\)y = \(2\). Collecting like terms gives \(2\).
  4. This equals \(2\), proving the identity on the stated domain.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate y=\(\operatorname{atan}^{2}{\left(x \right)}\) to obtain y′=\(\frac{2 \operatorname{atan}{\left(x \right)}}{x^{2} + 1}\).
  • [1 mark] Differentiate again, using the product and chain rules: y″=\(\frac{2 \left(- 2 x \operatorname{atan}{\left(x \right)} + 1\right)}{\left(x^{2} + 1\right)^{2}}\).
  • [1 mark] Substitute these derivatives into the left side \(\left(x^{2} + 1\right)^{2}\)y″ + \(2 x \left(x^{2} + 1\right)\)y′ + \(0\)y = \(2\). Collecting like terms gives \(2\).
  • [0.5 marks] This equals \(2\), proving the identity on the stated domain.

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