Problem 5) Like we saw above when reducing integrating factor ODEs to separable ODEs, many ODE techniques are reductions of an ODE we can't solve to one we can. We will look at a particular technique which is used to solve ODEs which we call Homogeneous. A homogeneous ODE is one in the following form (where g is some given function) Mm) = 9(9/ :13) (a) Introduce the variable to = y/x. Compute the derivative of w and solve for y'. (1)) Insert this into the homogeneous ODE (1:), and manipulate the result into a form you know how to solve. (c) What type of ODE do you now have? Solve this ODE and back substitute 9 = we: to obtain the solution to (at) (d) Solve the following homogeneous ODEs i. W: y? . i = 1 .. f w. m' sin(m)' u. y =log(y/a:). v. '=arctan(5/t). '13 9'2 '9 iii- PT?) = F. vi. (Hard): y' = a: (1+ 3) + E. This last one demonstrates that some manipulation is sometimes required to even reach the point of getting a homogeneous ODE