is the particular solution of the given differential equation. Example . Solve y dx x dy The given equation can be written as dx = This is a homogeneous equation. Let y vx .
Then we have v x dv = v v . Thus, x dv v v v or ) + v v dv x or − ) + v v v v dv x . Integrating both sides, we get − ) = tan v v v or log tan v v v ) = − or log tan v v v ) = − or log tan v v v ) = − Now replacing v by y x , we get, log tan k = , where k = − 2log gives the required solution. - - Ordinary Differential Equations Example .
Solve y x dy xy dy The given equation is rewritten as dy dx = This is a homogeneous differential equation. Put y vx . Then, we have x dv dx = v v − . By separating the variables, v v dv − = dx x .
Integrating, we obtain v v − log = log or v vxC = log Replacing v by y x , we get, y Cy = log or Cy e y x / or y ke y x / (how!) which is the required solution. Example . Solve y dy x y x y / / The given equation can be written as dx g x y x y / / …( ) The appearance of x y in equation ( ), suggests that the appropriate substitution is x vy Put x vy . Then, we have y dv v v v = − By separating the variables, we have = − v e dv v v On integration, we obtain e v v + = − or log ye vy v + or ye vy v + = ± Replace v by