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Class 12 Continuity And Differentiability · Long Answer3 Markssecond_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #93

For y=\(4 \sin{\left(\log{\left(x \right)} \right)} + 3 \cos{\left(\log{\left(x \right)} \right)}\), prove \(x^{2}\)y″ + \(x\)y′ + \(1\)y = \(0\). Assume x>0.

✅ Step-by-Step Solution

Final Answer: Identity proved.
  1. Differentiate y=\(4 \sin{\left(\log{\left(x \right)} \right)} + 3 \cos{\left(\log{\left(x \right)} \right)}\) to obtain y′=\(\frac{- 3 \sin{\left(\log{\left(x \right)} \right)} + 4 \cos{\left(\log{\left(x \right)} \right)}}{x}\).
  2. Differentiate again, using the product and chain rules: y″=\(- \frac{\sin{\left(\log{\left(x \right)} \right)} + 7 \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}\).
  3. Substitute these derivatives into the left side \(x^{2}\)y″ + \(x\)y′ + \(1\)y = \(0\). Collecting like terms gives \(0\).
  4. This equals \(0\), proving the identity on the stated domain.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Differentiate y=\(4 \sin{\left(\log{\left(x \right)} \right)} + 3 \cos{\left(\log{\left(x \right)} \right)}\) to obtain y′=\(\frac{- 3 \sin{\left(\log{\left(x \right)} \right)} + 4 \cos{\left(\log{\left(x \right)} \right)}}{x}\).
  • [1 mark] Differentiate again, using the product and chain rules: y″=\(- \frac{\sin{\left(\log{\left(x \right)} \right)} + 7 \cos{\left(\log{\left(x \right)} \right)}}{x^{2}}\).
  • [1 mark] Substitute these derivatives into the left side \(x^{2}\)y″ + \(x\)y′ + \(1\)y = \(0\). Collecting like terms gives \(0\).
  • [0.5 marks] This equals \(0\), proving the identity on the stated domain.

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