Miscellaneous

Chapter 9 Class 12 Differential Equations
Serial order wise

### Transcript

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 πππβ‘γπ+π πππβ‘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 πππβ‘γπ+π πππβ‘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) 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ππ^β²+ππ=π (π^2 π¦)/(ππ₯^2 )β2 ππ¦/ππ₯+2π¦=0 Thus, Given Function is a solution of the Differential Equation

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.