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Example 16 - Chapter 9 Class 12 Differential Equations (Term 2)

Last updated at March 16, 2023 by

Example 16 - Show x - cos (y/x) = y cos(y/x) + x is homogeneous - Solving homogeneous differential equation

Example 16 - Chapter 9 Class 12 Differential Equations - Part 2
Example 16 - Chapter 9 Class 12 Differential Equations - Part 3 Example 16 - Chapter 9 Class 12 Differential Equations - Part 4 Example 16 - Chapter 9 Class 12 Differential Equations - Part 5

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Transcript

Example 16 Show that the differential equation ( / )= ( / )+ is homogeneous and solve it. Step 1: Find / ( / ) / = cos ( / )+ / =( cos ( / ) + )/( cos ( / ) ) Step 2: Put F( , )= / & find F( , ) F( , )=( cos ( / ) + )/( cos ( / ) ) Finding F( , ) F( , )=(( ) ( / ) + )/(( ) . cos ( / ) ) =( ( / ) + )/( cos ( / ) ) = ( ( / ) + )/( cos ( / ) ) =( ( / ) + )/( cos ( / ) ) = F ( , ) So , F( , )= F( , ) = F( , ) Thus , F( , ) is a homogeneous function of degree zero. Therefore, the given differential equation is homogeneous differential equation Step 3: Solving / by Putting = / =( ( / ) + )/( cos ( / ) ) Put = So, / = ( ) = / . + / = / + Putting values of / and y = vx in (1) i.e. / = ( ( / )+ )/( cos ( / ) ) / + =(( ) ( / ) + )/( cos ( / ) ) / + =( ( ) + )/( cos ) / + = ( cos +1 )/( cos ) / + =( cos +1 )/cos / =( cos + 1 )/cos / =( cos + 1 cos )/cos / = 1/cos cos = / Integrating Both Sides 1 cos = 1 / sin =log | |+ 1 Putting = / & t 1=log / =log | |+log | | / = | |

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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.