Ex 9.4, 2 In each of the Exercise 1 to 10 , show that the given differential equation is homogeneous and solve each of them. 𝑦^′=(𝑥+𝑦)/𝑥
Step 1: Find 𝑑𝑦/𝑑𝑥
𝑑𝑦/𝑑𝑥 = (𝑥 + 𝑦)/𝑥
Step 2. Putting F(x, y) = 𝑑𝑦/𝑑𝑥 and find F(𝜆x, 𝜆y)
So, F(x, y) = (𝑥 + 𝑦)/𝑥
F(𝜆x, 𝜆y) = (𝜆𝑥 +𝜆𝑦)/𝜆𝑥 = (𝜆(𝑥 +𝑦))/𝜆𝑥 = (𝑥 + 𝑦)/𝑥 = F(x, y) = 𝜆°F(x, y)
Therefore F(x, y) Is a homogenous function of degree zero.
Hence 𝑑𝑦/𝑑𝑥 is a homogenous differential equation
Step 3: Solving 𝑑𝑦/𝑑𝑥 by putting y = vx
Put y = vx.
differentiating w.r.t.x
𝑑𝑦/𝑑𝑥 = x 𝑑𝑣/𝑑𝑥+𝑣𝑑𝑥/𝑑𝑥
𝑑𝑦/𝑑𝑥 = 𝑥 𝑑𝑣/𝑑𝑥 + v
Putting value of 𝑑𝑦/𝑑𝑥 and y = vx in (1)
𝑑𝑦/𝑑𝑥 = (𝑥 + 𝑦)/𝑥
𝑥 ( 𝑑𝑣)/𝑑𝑥 + v = (𝑥 + 𝑣𝑥)/𝑥
𝑥 ( 𝑑𝑣)/𝑑𝑥 + v = 1+𝑣
𝑥 (𝑥 𝑑𝑣)/𝑑𝑥 = 1+𝑣−𝑣
𝑥 ( 𝑑𝑣)/𝑑𝑥 = 1
( 𝑑𝑣)/𝑑𝑥 = 1/𝑥
Integrating both sides
∫1▒〖𝑑𝑣=∫1▒〖𝑑𝑥/𝑥 〗 〗
v = log|𝑥|+𝑐
Putting v = 𝑦/𝑥
𝑦/𝑥 = log|𝑥| + c
y = x log|𝒙| + cx

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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