Nonexact Differential Equation (Possible to solve by integrating factor?)

[TranscendArcu]

Homework Statement

Solve the differential equation: t^2 y' + y^2 = 0

The Attempt at a Solution

Now, it's definitely possible to solve this via separable of variables. But I am curious to know if I can solve it with an integrating factor. Having done some reading, I noticed that this equation is nearly in the form of an exact differential. Rewriting:

t^2 y' + y^2 = 0 = t^2 \frac{dy}{dt} + y^2, implies,
t^2 dy + y^2 dt = 0 = y^2 dt + t^2 dy.

Unfortunately, letting M(x,y) = y^2 and N(x,y) = t^2 and then taking derivatives shows M_y = 2y ≠ N_t = 2t, so it looks like an exact equation isn't going to emerge from this.

In the event that the equation is not exact, an integrating factor is typically sought. The problem is, I don't know how to go about finding such an thing. Can someone help me?

 

[hikaru1221]

You can try with this: \frac{dy}{y^2} + \frac{dt}{t^2} = 0

 

[LCKurtz]

Homework Statement

Solve the differential equation: t^2 y' + y^2 = 0

The Attempt at a Solution

Now, it's definitely possible to solve this via separable of variables.

You can try with this: \frac{dy}{y^2} + \frac{dt}{t^2} = 0

I think he know that.
@TranscendArcu: It is not a given that a given first order DE can be solved by an integrating factor in any practical fashion, even if you can solve it by separation of variables.