Let dy/dx = f(x,y). If f(x, y) can be expressed as a function of the ratio...

 

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Let dy/dx = f(x,y). If f(x, y) can be expressed as a function of the ratio y/x only, then the differential equation is said to be homogeneous. dy 4x + y dx x - 4y (a) (1 mark) Show that (b) (2 marks) It turns out that homogeneous equations can always be transformed into separable equations by a change of the dependent variable in the following way: Let y = xv. Transform the differential equation in (a) into an equation in v and x. Show that the equation is separable. (c) (3 marks) Show that the solution of the differential equation in (a) is 1 = is homogeneous. arctan-In (a + y) = ( C. X (d) (2 marks) Express the equation in (c) in terms of polar coordinates. (e) (2 marks) Use (d) to sketch a solution curve for the differential equation in (a). Let dy/dx = f(x,y). If f(x, y) can be expressed as a function of the ratio y/x only, then the differential equation is said to be homogeneous. dy 4x + y dx x - 4y (a) (1 mark) Show that (b) (2 marks) It turns out that homogeneous equations can always be transformed into separable equations by a change of the dependent variable in the following way: Let y = xv. Transform the differential equation in (a) into an equation in v and x. Show that the equation is separable. (c) (3 marks) Show that the solution of the differential equation in (a) is 1 = is homogeneous. arctan-In (a + y) = ( C. X (d) (2 marks) Express the equation in (c) in terms of polar coordinates. (e) (2 marks) Use (d) to sketch a solution curve for the differential equation in (a).