Miscellaneous

Chapter 9 Class 12 Differential Equations
Serial order wise

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Misc 2 For each of the exercise given below , verify that the given function (ππππππππ‘ ππ ππ₯ππππππ‘) is a solution of the corresponding differential equation . (ii) π¦=π^π₯ (π cosβ‘γπ₯+π sinβ‘π₯ γ ) : (π^2 π¦)/(ππ₯^2 )β2 ππ¦/ππ₯+2π¦=0 π¦=π^π₯ (π cosβ‘γπ₯+π sinβ‘π₯ γ ) : Differentiating w.r.t. x π¦^β²=[π^π₯ (π cosβ‘γπ₯+π sinβ‘π₯ γ )]^β² π¦^β²=γ(πγ^π₯)β² (π cosβ‘γπ₯+π sinβ‘π₯ γ )+π^π₯ (π cosβ‘γπ₯+π sinβ‘π₯ γ )β² π¦^β²=π^π (π πππβ‘γπ+π πππβ‘π γ )+π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ ) Putting π¦=π^π₯ (π πππ β‘γπ₯+π π ππβ‘π₯ γ ) : π¦^β²=π+π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ ) π¦^β²βπ¦=π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ ) Differentiating again w.r.t x π¦^β²β²βπ¦^β²=[π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ )]^β² π¦^β²β²βπ¦^β²=γ(πγ^π₯)β²(βπ sinβ‘γπ₯+π cosβ‘π₯ γ )+π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ )β² π¦^β²β²βπ¦^β²=π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ )+π^π₯ (βπ cosβ‘γπ₯βπ siπβ‘π₯ γ ) π¦^β²β²βπ¦^β²=π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ )βπ^π₯ (π cosβ‘γπ₯+π siπβ‘π₯ γ ) Putting π¦=π^π₯ (π πππ β‘γπ₯+π π ππβ‘π₯ γ ) π¦^β²β²βπ¦^β²=π^π₯ (βπ sinβ‘γπ₯+π cosβ‘π₯ γ )βπ¦ Putting π¦^β²βπ¦=π^π₯ (βπ π ππβ‘γπ₯+π πππ β‘π₯ γ ) from (1) β¦(1) π¦^β²β²βπ¦^β²=π¦^β²βπ¦βπ¦ π¦^β²β²βπ¦^β²=π¦^β²β2π¦ π¦^β²β²βπ¦^β²βπ¦^β²+2π¦=0 π¦^β²β²β2π¦^β²+2π¦=0 (π^2 π¦)/(ππ₯^2 )β2 ππ¦/ππ₯+2π¦=0 Thus, Given Function is a solution of the Differential Equation