Class 12 Continuity And Differentiability · Long Answer3 Markssecond_derivatives

Class 12 Maths · Continuity And Differentiability · Long Answer #91

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

✅ Step-by-Step Solution

Final Answer: Identity proved.

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

[0.5 marks] Differentiate y=\(\sin{\left(\log{\left(x \right)} \right)}\) to obtain y′=\(\frac{\cos{\left(\log{\left(x \right)} \right)}}{x}\).
[1 mark] Differentiate again, using the product and chain rules: y″=\(- \frac{\sqrt{2} \sin{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{x^{2}}\).
[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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