# Misc 7

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 7 Show that the general solution of the differential equation 𝑑𝑦𝑑𝑥+ 𝑦2+𝑦+1 𝑥2+𝑥+1=0 is given by 𝑥+𝑦+1=A 1−𝑥−𝑦−2𝑥𝑦, where A is parameter. 𝑑𝑦𝑑𝑥+ 𝑦2 + 𝑦 + 1 𝑥2 + 𝑥 + 1 = 0 𝑑𝑦𝑑𝑥= −( 𝑦2 + 𝑦 + 1) 𝑥2 + 𝑥 + 1 𝑑𝑦 𝑦2 + 𝑦 + 1= −𝑑𝑥 𝑥2 + 𝑥 + 1 𝑑𝑦𝑦 +2 12𝑦 + 122− 122+ 1= −𝑑𝑥𝑦 +2 12𝑦 + 122− 122+ 1 𝑑𝑦 𝑦 + 122+ 34= −𝑑𝑥 𝑥 + 122+ 34 𝑑𝑦 𝑦 + 122+ 322= −𝑑𝑥 𝑥 + 122+ 322 Integrating both sides 𝑑𝑦 𝑦 + 122 + 322 = − 𝑑𝑥 𝑥 + 122 + 322 2 3 tan−1 𝑦 + 12 32 = −2 3 tan−1 𝑥 + 12 32 2 3 tan−1 2𝑦 + 1 3 + tan−1 2𝑥 + 1 3 = C 2 3 tan−1 2𝑦 + 1 3+ 2𝑥 + 1 31− 2𝑦 − 1 3 × 2𝑥 + 1 3 =𝐶 2𝑦 + 1 + 2𝑥 + 1 31− (2𝑦 + 1)(2𝑥 + 1) 3 = tan 32𝐶 2 3 tan−1 2𝑦 + 12 32 = − 2 3 tan−1 2𝑥 + 12 32 = C 2 3 tan−1 2𝑦 + 1 3 + tan−1 2𝑥 + 1 3 = C 2 3 tan−1 2𝑦 + 1 3 + 2𝑥 + 1 31 − 2𝑦 − 1 3 × 2𝑥 + 1 3 =𝐶 2𝑦 + 1 + 2𝑥 + 1 31 − (2𝑦 + 1)(2𝑥 + 1)3 = tan 32𝐶 2𝑦 + 1 + 2𝑥 + 1 3 3 − (2𝑦 + 1)(2𝑥 + 1)3 = C1 3(2𝑦 + 2𝑥 + 2)3 − 4𝑥𝑦 + 2𝑦 + 2𝑥 + 1 = C 2 3 (x + y + 1) = C1 3−4𝑥𝑦−2𝑥−2𝑦−1 2 3 (x + y + 1) = C1 2−4𝑥𝑦−2𝑥−2𝑦 𝟑 (x + y + 1) = C1 ( 1 − x − y − 2xy) is the required general solution

Chapter 9 Class 12 Differential Equations

Serial order wise

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.