# Ex 9.5, 15

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Ex 9.5, 15 For each of the differential equations in Exercises from 11 to 15 , find the particular solution satisfying the given condition : 2𝑥𝑦+ 𝑦2−2 𝑥2 𝑑𝑦𝑑𝑥=0;𝑦=2 When 𝑥=1 Differential equation can be written 𝑎s 2𝑥𝑦+ 𝑦2−2 𝑥2 𝑑𝑦𝑑𝑥=0 2 𝑥2 𝑑𝑦𝑑𝑥=2𝑥𝑦+ 𝑦2 𝑑𝑦𝑑𝑥= 2𝑥𝑦 + 𝑦22 𝑥2 𝑑𝑦𝑑𝑥= 𝑦𝑥 + 𝑦22 𝑥2 Let F(x, y) = 𝑑𝑦𝑑𝑥 = 𝑦𝑥 + 𝑦22 𝑥2 Finding F(𝜆x, 𝜆y) F(𝜆x, 𝜆y) = 𝜆𝑦𝜆𝑥 + (𝜆𝑦)22 (𝜆𝑥)2 = 𝑦𝑥 + 𝑦22 𝑥2 = 𝜆° F(x, y) ∴ F(x, y) is a homogenous function of degree zero Putting y = vx Diff w.r.t. x 𝑑𝑦𝑑𝑥 = x 𝑑𝑣𝑑𝑥 + v Putting value of 𝑑𝑦𝑑𝑥 and y = vx in (1) 𝑑𝑦𝑑𝑥= 𝑦𝑥 + 𝑦22 𝑥2 𝑣+𝑥 𝑑𝑣𝑑𝑥 = 𝑣𝑥𝑥 + 12 𝑣2 𝑥2 𝑥2 𝑣+𝑥 𝑑𝑣𝑑𝑥 = 𝑣+ 𝑣22 𝑥𝑑𝑣𝑑𝑥 = 𝑣 + 𝑣22 − v 𝑥𝑑𝑣𝑑𝑥 = 𝑣22 2𝑑𝑣 𝑣2 = 𝑑𝑥𝑥 Integrating both sides 2 𝑑𝑣 𝑣2 = 𝑓 𝑑𝑥𝑥 2 𝑣−2 𝑑𝑣= log 𝑥+𝑐 2 𝑣−2 + 1 −2 + 1 = log 𝑥+𝑐 2 𝑉 − 1 −1 = log 𝑥+𝑐 −2 𝑣 = log 𝑥+𝑐 Putting value of v = 𝑦 𝑥 −2𝑥𝑦 = log 𝑥 + C Put x = 1 & y = 2 −2(1)2 = log 1 + C − 1 = 0 + C C = −1 Putting value in (2) −2𝑥𝑦 = log 𝑥 − 1 y = −2𝑥 log 𝑥 − 1 y = 𝟐𝒙 𝟏 − 𝐥𝐨𝐠 𝒙

Example 1
Important

Ex 9.1, 11 Important

Ex 9.1, 12 Important

Example 7 Important

Ex 9.3, 7 Important

Ex 9.3, 10 Important

Example 13 Important

Ex 9.4, 14 Important

Example 17 Important

Example 18 Important

Ex 9.5, 8 Important

Ex 9.5, 15 Important You are here

Example 22 Important

Ex 9.6, 7 Important

Ex 9.6, 13 Important

Ex 9.6, 14 Important

Example 25 Important

Example 27 Important

Example 28 Important

Misc 6 Important

Misc 11 Important

Misc 12 Important

Misc 13 Important

Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.