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

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  1. Chapter 9 Class 12 Differential Equations
  2. Serial order wise
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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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