Ex 9.5, 9 - Chapter 9 Class 12 Differential Equations

Transcript

Ex 9.5, 9 In each of the Exercise 1 to 10 , show that the given differential equation is homogeneous and solve each of them. + 2 =0 Step 1 : Find + 2 =0 dy log 2 = y dx = log 2 = 2 log = 2 log Step 2. Putting F(x , y) = and finding F( x, y) F(x, y) = 2 log ( , ) = 2 log = 2 log = ( , ) Thus, F(x, y) is a homogenous equation function of order zero Therefore is a homogenous differential equation Step 3 : Solving by putting y = vx Putting y = vx Diff w.r.t.x = x + v = x + v Putting value of and y = vx in (1) = 2 log v + = 2 v + = 2 log = 2 log v = 2 + log 2 log = log 2 log 2 log log dv = Integrating both sides 2 log log dv = 2 log (1 log ) dv = log x + log c 1 + 1 log (1 log ) = log x + log c 1 ( )(1 log ) 1 = log x + log c 1 ( log 1) 1 = log x + log c ( log 1) log v = log x + log c Put t = log v 1 dt = 1 dv So, our equation becomes log v = log x + log c log t log v = log x + log c Putting value of t log (log v 1) log v + = log x + log c log (log v 1) = log x + log c + log v log (log v 1) = log C xv Putting value of v = log log 1 = log log log 1 = log log 1 = cy cy = log 1