What is the uniqueness of solutions for linear equations at x=0?

[wachaif]

(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)= yo has a unique solution when yo = 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.

 

[saltydog]

How about (a):

Set it up in std. form for first order ODE, calc integrating factor, solve. You get:

$$y(x)=x+\frac{y_0}{x^2}$$

Well, the only way for this to have a solution in the Reals for initial value problem is for $y_0=0$ in which case, solution is $y=x$. However, if y(0)$\neq 0$, then no solution exists at x=0 since this is indeterminate.

Is there a more rigorous way to say this?

 

[Jayboy]

and (b) gives:

(...i'll spare the details but it's the usual integrating factor prob...)

$$y = x(cx - 3)$$

where c is your integration constant.

To satisfy the IC y(0) = 0, c can take on any value you want.