Question

Solution of (d²y/dx²) = log x is: 

(A) y = (1/2)x² log x – (3/4)x² + c₁x + c₂

(B) y = (1/2)x² log x + (3/4)x² + c₁x + c₂

(C) y = (1/2)x² log x – (3/4)x² – c₁x + c₂ 

(D) None of these

Answer 1

The correct option (A) y = (1/2)x² log x – (3/4)x² + c₁x + c₂

Explanation:

(d²y/dx²) = log x 

∴ (dy/dx) = ∫log x ∙ dx 

(dy/dx) = x log x – x + c 

∴ y = ∫x log x dx – ∫x dx + ∫c ∙ dx 

Consider 

∫x ∙ log x dx = log x∫x dx – ∫(x²/2) × (1/x)dx 

= log x ∙ (x²/2) – (x²/4) 

∴ y = log x ∙ (x²/2) – (x²/4) – (x²/2) + cx + d 

= (1/2)x² log x – (3/4)x² + cx + d 

∴ y = (1/2)x² log x – (3/4)x² + c₁x + c₂