Last updated at Aug. 20, 2021 by Teachoo

Transcript

Misc 2 For each of the exercise given below , verify that the given function (๐๐๐๐๐๐๐๐ก ๐๐ ๐๐ฅ๐๐๐๐๐๐ก) is a solution of the corresponding differential equation . (i) ๐ฅ๐ฆ=๐ ๐^๐ฅ+๐ ๐^(โ๐ฅ)+๐ฅ^2 : ๐ฅ (๐^2 ๐ฆ)/(๐๐ฅ^2 )+2 ๐๐ฆ/๐๐ฅโ๐ฅ๐ฆ+๐ฅ^2โ2=0 ๐ฅ๐ฆ=๐๐^๐ฅ+๐ ๐^(โ๐ฅ)+๐ฅ^2 Differentiating w.r.t x (๐(๐ฅ๐ฆ))/๐๐ฅ=๐/๐๐ฅ [๐ ๐^๐ฅ+๐ ๐^(โ๐ฅ)+๐ฅ^2 ] ๐๐ฅ/๐๐ฅ y+๐๐ฆ/๐๐ฅ ๐ฅ =๐ใ ๐ใ^๐ฅ+(โ1)๐ ๐^(โ๐ฅ)+2๐ฅ y+y^โฒ x =aใ eใ^xโb e^(โx)+2x Differentiating again w.r.t x ๐ฆโฒ+(๐ฆ^โฒ ๐ฅ)^โฒ =(๐๐^๐ฅ )^โฒโ(๐๐^(โ๐ฅ) )^โฒ+(2๐ฅ)^โฒ ๐ฆ^โฒ+(๐ฆ^โฒโฒ ๐ฅ+๐ฆ^โฒร1)=๐๐^๐ฅ+๐๐^(โ๐ฅ)+2 ๐ฆ^โฒโฒ ๐ฅ+2๐ฆ^โฒ=๐๐^๐ฅ+๐๐^(โ๐ฅ)+2 Now, we know that ๐ฅ๐ฆ=๐๐^๐ฅ+๐ ๐^(โ๐ฅ)+๐ฅ^2 ๐ฅ๐ฆโ๐ฅ^2=๐๐^๐ฅ+๐ ๐^(โ๐ฅ) ๐๐^๐ฅ+๐ ๐^(โ๐ฅ)=๐ฅ๐ฆโ๐ฅ^2 ...(1) (Given equation) ...(2) Putting (2) in (1) ๐ฆ^โฒโฒ ๐ฅ+2๐ฆ^โฒ=๐๐^๐+๐๐^(โ๐)+2 ๐ฆ^โฒโฒ ๐ฅ+2๐ฆ^โฒ=๐๐โ๐^๐+2 (๐^2 ๐ฆ)/(๐๐ฅ^2 ) ๐ฅ+2 ๐๐ฆ/๐๐ฅ=๐๐โ๐^๐+2 (๐ ^๐ ๐)/(๐ ๐^๐ ) ๐+๐ ๐ ๐/๐ ๐โ๐๐+๐^๐=๐ โด The given function is a solution

Miscellaneous

Misc 1 (i)

Misc 1 (ii)

Misc 1 (iii) Important

Misc 2 (i) You are here

Misc 2 (ii) Important

Misc 2 (iii)

Misc 2 (iv) Important

Misc 3 Deleted for CBSE Board 2022 Exams

Misc 4 Important

Misc 5 Important Deleted for CBSE Board 2022 Exams

Misc 6

Misc 7 Important

Misc 8

Misc 9 Important

Misc 10 Important

Misc 11

Misc 12 Important

Misc 13

Misc 14 Important

Misc 15 Important

Misc 16 (MCQ)

Misc 17 (MCQ) Important Deleted for CBSE Board 2022 Exams

Misc 18 (MCQ)

Chapter 9 Class 12 Differential Equations (Term 2)

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.