Check sibling questions

Ex 9.5, 4 - Show homogeneous: (x2 - y2) dx + 2xy dy = 0 - Ex 9.5

Ex 9.5, 4 - Chapter 9 Class 12 Differential Equations - Part 2
Ex 9.5, 4 - Chapter 9 Class 12 Differential Equations - Part 3 Ex 9.5, 4 - Chapter 9 Class 12 Differential Equations - Part 4 Ex 9.5, 4 - Chapter 9 Class 12 Differential Equations - Part 5 Ex 9.5, 4 - Chapter 9 Class 12 Differential Equations - Part 6

Maths Crash Course - Live lectures + all videos + Real time Doubt solving!


Transcript

Ex 9.5, 4 show that the given differential equation is homogeneous and solve each of them. ( ^2 ^2 ) +2 =0 Step 1: Find / ( ^2 ^2 ) +2 =0 2xy dy = ( ^2 ^2 ) dx 2xy dy = ( ^2 ^2 ) dx / = ( ^2 ^2)/2 Step 2: Putting F(x, y) = / and finding F( x, y) F(x, y) = ( ^2 ^2)/2 F( x, y) = (( )^2 ( )^2)/(2 . )= ( ^2 ^2 ^2 ^2)/( ^2.2 )= ( ^2 ( ^2 ^2))/( ^2.2 ) = ( ^2 ^2)/2 = F(x, y) F( x, y) = F(x, y) = F(x, y) Hence, F(x, y) is a homogenous Function of with degree zero So, / is a homogenous differential equation. Step 3. Solving / by putting y = vx Put y = vx. differentiating w.r.t.x / = x / + / / = / + v Putting value of / and y = vx in (1) / = ( ^2 ^2)/2 ( )/ + = ( ( ) ^2 ^2)/(2 ( )) ( )/ + = ( ^2 ^2 ^2)/(2 ^2 ) ( )/ = ( ^2 ^2 ^2)/(2 ^2 ) v x / = ( ^2 ^2 ^2 2 ^2 ^2)/(2 ^2 ) x / = ( ^2 ^2 ^2)/(2 ^2 ) / = 1/ (( ^2 ( ^2 + 1))/(2 ^2 )) / = 1/ (( ^2 + 1)/2 ) (2 )/( ^2 + 1) = ( )/ Integrating both sides 1 2 /( ^2 + 1) = 1 ( )/ 1 2 /( ^2 + 1) = log | |+ Putting t = v2 + 1 diff.w.r.t v. / (v2 + 1) = / 2v = / dv = /2 Now, From (2) 1 2 / /2 = log| |+ ( from (2) ) 1 / = log| |+ log| | = log| |+ Putting t = v2 + 1 log|"v2 + 1" | = log|"x" |+ log |"v2 + 1" | + log|"x" |= log |"x(v2 + 1)" | = C Putting v = / log |[( / )^2+1] |= log|( ^2 + ^2)/ ^2 |= Putting = log c log |( ^2 + ^2)/ | = log c ^2+ ^2 = cx ^ + ^ = is the general solution of the given differential equation.

Ask a doubt (live)
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.