1. Find the general solution to the equations/initial-value problems below (10 points each). (a) y' t xy = 6xvy (b) y(-1) =2 2. We will see different usages of the term homogeneous equation. For the purposes of this assignment we will define a homogeneous equation as follows. Consider the equation dy dr - = f(I, y). If we can write f(r, y) as a function of >, then we say the equation is ho- mogeneous, and we can solve it as a separable equation. For this problem, we will specifically consider the equation dy 2xy 12 - (a) (3 points) Show that we can write the equation above as dy 24 dx 1 - (4) 2 (b) (5 points) We now use the trick for solving these types of homoge- neous equations: define v(x) := zy(x). Write the equation above in terms of r and the unknown function v. (There should be no y in your equation.) (c) (12 points) Solve the equation from part (c) for v. You may leave your solution for v expressed implicitly (that is, you do not need to isolate v). (d) (5 points) Determine a solution for y. 3. Determine if each of the equations below is an exact equation. If so, solve the differential equation/initial value problem (10 points each). (a) 12 + 12 + (2tx - I) dx = 0 (b) dy sin a cosy - sin y cos " d -=0 (c) Jy+ 2xy3 + (1 +3x7y) + x)12 (y(1) = -5