Chapter 5 Class 12 Continuity and Differentiability

Class 12
Important Questions for exams Class 12

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

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Ex 5.7, 14 If π¦= γAπγ^ππ₯ + γBπγ^ππ₯, show that π2π¦/ππ₯2 β (π+π) ππ¦/ππ₯ + πππ¦ = 0 π¦= γAπγ^ππ₯ + γBπγ^ππ₯ Differentiating π€.π.π‘.π₯ ππ¦/ππ₯ = (π(γAπγ^ππ₯ " + " γBπγ^ππ₯))/ππ₯ ππ¦/ππ₯ = (π(γAπγ^ππ₯))/ππ₯ + (π(γBπγ^ππ₯))/ππ₯ ππ¦/ππ₯ = A . π^ππ₯. (π(ππ₯))/ππ₯ + B . π^ππ₯ (π(ππ₯))/ππ₯ ππ¦/ππ₯ = A . π^ππ₯. π + B . π^ππ₯. π ππ¦/ππ₯ = π΄ππ^ππ₯ + π΅ππ^ππ₯ Again Differentiating π€.π.π‘.π₯ π/ππ₯ (ππ¦/ππ₯) = π(π΄ππ^ππ₯ " + " π΅ππ^ππ₯ " " )" " /ππ₯ (π^2 π¦)/(ππ₯^2 ) = π(π΄ππ^ππ₯ )/ππ₯ + π(π΅ππ^ππ₯ )" " /ππ₯ (π^2 π¦)/(ππ₯^2 ) = π΄π π(π^ππ₯ )/ππ₯ + π΅π π(π^ππ₯ )" " /ππ₯ (π^2 π¦)/(ππ₯^2 ) = π΄π . π^(ππ₯ ). π(ππ₯ )/ππ₯ + π΅π . π^ππ₯ . π(ππ₯)/ππ₯ (π^2 π¦)/(ππ₯^2 ) = π΄ππ^(ππ₯ ) . π+π΅ππ^ππ₯ . π (π^2 π¦)/(ππ₯^2 ) = π΄π2π^(ππ₯ )+π΅π2π^ππ₯ We need to prove (π^2 π¦)/(ππ₯^2 ) β (π+π) ππ¦/ππ₯ + πππ¦ = 0 Solving LHS (π^2 π¦)/(ππ₯^2 ) β (π+π) ππ¦/ππ₯ + πππ¦ = (π΄π2π^(ππ₯ )+π΅π2π^ππ₯) β (π+π) (π΄ππ^(ππ₯ )+π΅ππ^ππ₯) + ππ (π΄π^(ππ₯ )+π΅π^ππ₯) = π΄π2π^(ππ₯ )+π΅π2π^ππ₯ β π(π΄ππ^(ππ₯ )+π΅ππ^ππ₯) β π(π΄ππ^(ππ₯ )+π΅ππ^ππ₯) + ππ π΄π^(ππ₯ )+πππ΅π^ππ₯ = π΄π2π^(ππ₯ )+π΅π2π^ππ₯ βπ΄π2π^(ππ₯ )β π΅πππ^ππ₯ β π΄πππ^(ππ₯ ) + π΅π2π^ππ₯+ ππ π΄π^(ππ₯ )+πππ΅π^ππ₯ = π΄π2π^(ππ₯ )β π΄π2π^(ππ₯ ) + π΅π2π^ππ₯ βπ΅π2π^ππ₯ β π΅πππ^ππ₯ + π΅πππ^ππ₯ β π΄πππ^(ππ₯ ) + π΄πππ^(ππ₯ ) = 0 = RHS Hence proved