Differential Equations

Rule for solving homogeneous equation Let the given equation be homogeneous. Then, by definition, the given equation can be put in the form dy/dx = ƒ(y/x) ---------- (1) Solution Method: - To solve (1), let y/x = v or, y = vx Differentiating equation (1) with respect to x, we get dy/dx = v + x (dv/dx)----------- (2) So the equation becomes v + x (dv/dx) = ƒ (v) or, x (dv/dx) = ƒ (v) – v Separating the variables x and v, we have dx/x = dv/{ƒ (v) – v}--------------(3) Integrating equation (3) After integration, replace v by y/x and finally we will find the required solution. Example: Solve: (x 3 + 3xy 2 )dx + (y 3 + 3x 2 y)dy = 0 Given that, (x 3 + 3xy 2 )dx + (y 3 + 3x 2 y)dy = 0 or, dy/dx = - {(x 3 + 3xy 2 )/ (y 3 + 3x 2 y)} or, dy/dx = - [{1 + 3(y/x) 2 }/{(y/x) 3 + 3(y/x)}] -------(1) Take, y/x = v or, y = vx Differentiating with respect to x, we get dy/dx = v + x (dv/dx)----------- (2) From equation (1) and (2) v + x (dv/dx)