hi yaz!

i'm having some difficulty understanding the "existence and uniqueness of solution" theorem. if someone could be so nice as to work this problem out and explain it to me.

1)a. show that xy' + 2y = 3x has only one solution defined at x=0. then show that the initial value problem for this equation with initial condition y(0) = y0 has a unique solution when y0 = 0 and no solution when y0≠0.

b. show that xy'-2y=3x has an infinite number of solutions defined at x=0. then show that the initial value problem for this equation with initial condition y(0)=0 has an infinite number of solutions.

also this other problem has me stumped!!!

dy/dx + P(x)y = x, y(0) = 1

where P(x) = {1, 0≤x≤2

{3, x>2

1)general solution for 0≤x≤2 is y=x-1+ce^(-x) and c=2

2)general solution for x>2 is y1/3)x-(1/9)+ce^(-3x)

question is: choose the constant in the general solution from part 2 so that the solution from part 1 and the solution from part 2 agree at x=2.