# Example 18 - Chapter 9 Class 12 Differential Equations (Term 2)

Last updated at Aug. 20, 2021 by Teachoo

Last updated at Aug. 20, 2021 by Teachoo

Transcript

Example 18 Show that the family of curves for which the slope of the tangent at any point , on its 2 + 2 2 , is given by 2 2 = We know that the slope of the tangent at , of a curve is Given slope of tangent at , is 2 + 2 2 Therefore = 2 + 2 2 Step 1: Find = 2 + 2 2 Step 2: Put = F So, F = 2 + 2 2 Finding F , F , = 2 + 2 2 = 2 2 + 2 2 2 2 = 2 2 + 2 2 2 = 2 + 2 2 = F , So , F , = F , = F , So , F , is homogeneous function of degree zero, Therefore given equation is a homogeneous differential equation Step 3: Solving by putting = = 2 + 2 2 Put = Diff. w.r.t. = = + = + Putting values and y in (i) v + = 2 + ( ) 2 2 ( ) v + = 2 + 2 2 2 2 v + = 2 + 2 2 2 2 . 2 2 v + = 2 + 2 2 2 2 2 2 2 v + = 2 + 2 2 2 2 2 2 2 = 2 + 2 2 2 2 2 2 2 = 2 2 2 2 2 = 2 1 2 2 . 2 = 1 2 2 = 1 2 2 . 1 2 1 2 = 2 2 1 = 2 2 1 = Integrating Both Sides 2 2 1 = 2 2 1 = 2 2 1 = + Solving Put 2 1= Diff. w.r.t. 2 1 = 2 = = 2 2 2 1 = 2 2 = = log Putting t = v2 1 = log 2 1 From (2) 2 2 1 = + log 2 1 = + 1 Putting = or = log 2 1 = + 1 log 2 1 = + 1 log 2 1 + =+ 1 2 1 = 1 2 2 1 = 1 Putting 1= log 2 2 1 = log 1 Removing log 2 2 1 = 1 2 2 = 1 2 = 1 2 2 = 1 2 2 = 1 2 2 = 1 Put = 1 = Hence Proved

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Chapter 9 Class 12 Differential Equations (Term 2)

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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.