CBSE Class 12 Sample Paper for 2019 Boards

Class 12
Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Question 17

If y = cos (m cos -1 ⁡x)

Show that : (1 - x 2 ) d 2 y/dx 2 – x dy/dx + m 2 y = 0

### Transcript

Question 17 If y = cosβ‘γ(π cos^(β1)β‘π₯)γ Show that : (1βπ₯^2) (π^2 π¦)/(ππ₯^2 ) β π₯ ππ¦/ππ₯ + π^2 π¦ = 0 y = cosβ‘γ(π cos^(β1)β‘π₯)γ Differentiating w.r.t. x ππ¦/ππ₯ = π(cosβ‘(π cos^(β1)β‘π₯ ) )/ππ₯ ππ¦/ππ₯ = sinβ‘(π cos^(β1)β‘π₯ ) π(π cos^(β1)β‘π₯ )/ππ₯ ππ¦/ππ₯ = sinβ‘(π cos^(β1)β‘π₯ )Γπ π(cos^(β1)β‘π₯ )/ππ₯ ππ¦/ππ₯ = sinβ‘(π cos^(β1)β‘π₯ )ΓπΓ(β1)/β(1 β π₯^2 ) ππ¦/ππ₯ = βm sinβ‘(π cos^(β1)β‘π₯ )Γ1/β(1 β π₯^2 ) π¦β² = βm sinβ‘(π cos^(β1)β‘π₯ )Γ1/β(1 β π₯^2 ) Now, finding π¦β²β² will be complicated So, we multiply β(π β π^π ) to the left side β(1βπ₯^2 ) π¦β² = βm sinβ‘(π cos^(β1)β‘π₯ ) Squaring both sides (β(1βπ₯^2 ) )^2 (π¦^β² )^2 = (βm)^2 sin^2β‘(π cos^(β1)β‘π₯ ) ππ¦/ππ₯ = sinβ‘(π cos^(β1)β‘π₯ )ΓπΓ(β1)/β(1 β π₯^2 ) ππ¦/ππ₯ = βm sinβ‘(π cos^(β1)β‘π₯ )Γ1/β(1 β π₯^2 ) π¦β² = βm sinβ‘(π cos^(β1)β‘π₯ )Γ1/β(1 β π₯^2 ) Now, finding π¦β²β² will be complicated So, we multiply β(π β π^π ) to the left side β(1βπ₯^2 ) π¦β² = βm sinβ‘(π cos^(β1)β‘π₯ ) Squaring both sides (β(1βπ₯^2 ) )^2 (π¦^β² )^2 = (βm)^2 sin^2β‘(π cos^(β1)β‘π₯ ) ππ¦/ππ₯ = sinβ‘(π cos^(β1)β‘π₯ )ΓπΓ(β1)/β(1 β π₯^2 ) ππ¦/ππ₯ = βm sinβ‘(π cos^(β1)β‘π₯ )Γ1/β(1 β π₯^2 ) π¦β² = βm sinβ‘(π cos^(β1)β‘π₯ )Γ1/β(1 β π₯^2 ) Now, finding π¦β²β² will be complicated So, we multiply β(π β π^π ) to the left side β(1βπ₯^2 ) π¦β² = βm sinβ‘(π cos^(β1)β‘π₯ ) Squaring both sides (β(1βπ₯^2 ) )^2 (π¦^β² )^2 = (βm)^2 sin^2β‘(π cos^(β1)β‘π₯ ) (1βx^2 ) (π¦^β² )^2 = π^2 sin^2β‘(π cos^(β1)β‘π₯ ) We know that sin2 x = 1 β cos2 x (1βx^2 ) (π¦^β² )^2 = π^2 (1βcos^2β‘(π cos^(β1)β‘π₯ ) ) Putting y = cosβ‘γ(π cos^(β1)β‘π₯)γ (1βx^2 ) (π¦^β² )^2 = π^2 (1βπ¦^2 ) Differentiating again w.r.t. x π(1 β x^2 )/ππ₯ (π¦^β² )^2 + (1βx^2 ) (π(π¦^β² )^2)/ππ₯ = π^2 π(1 β π¦^2 )/ππ₯ β2x (π¦^β² )^2 + (1βx^2 )2π¦^β² π¦β²β² = π^2Γβ2π¦π¦β² 2π¦^β² (βπ₯π¦^β² "+ " (1βx^2 )π¦β²β²) = β2y^β²Γπ^2 π¦ βπ₯π¦^β² "+ " (1βx^2 )π¦β²β² = βπ^2 π¦ (1βx^2 )π¦β²β² β π₯π¦^β² + π^2 π¦ = 0 (1βπ₯^2) (π^2 π¦)/(ππ₯^2 ) β π₯ ππ¦/ππ₯ + π^2 π¦ = 0 Hence proved