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# Differential Equations/A Ricatti equation

Question
Hi,
Consider the following IVP:
y'=y^2+a(x)y+b(x)  y(x0)=y0
where the functions a and b are continuous on the open interval (a,b) that contains x0.
As you surely know, Picard's existence theorem guarantees existence and uniqueness over an interval I centered at x0 and is a subInterval of (a,b), but the size of the interval is dependent on the maximum of f(x,y)=y^2+a(x)y+b(x).
so can you help me to find the broadest existence and uniqueness (or at least just existence) interval depending on Picard's theorem (or any other one) in terms of the functions a and b.

Sincere thanks for your time and effort.

The quadratic equation would say that y = (-a(x)±sqrt[a²(x)-4b(x)])/2.
This says that two solutions exist where a²(x) is greater than 4b(x) and
that only one solution exists where a²(x) = 4b(x).

The equation a²(x) = 4b(x) has a possibility of having 0, 1, or 2 solutions for x.
To get any more depth on the solution, we need to know something about a(x) and b(x).

For example, are they continuous?  Are they always positive?
Do they oscillate between a max and min?

Differential Equations

Volunteer

#### Scott A Wilson

##### Expertise

I am capable of solving most ordinary differential equations and a few not so ordinary, but those are few and far between.

##### Experience

I have taken Differential Equations at OSU. Occasionally I have assisted people with questions on the subject.

Publications
I wrote a thesis on the study of shock waves and rarefaction fans. That occurs in nature when mathematics fails to apply. See, mathematics says that when shock waves appear, there should be two solutions whereas nature knows there is only one way to be in that area. When rarefaction fans occur, the mathematics says there should be no solution, but despite this, nature still moves ahead with its own plan.

Education/Credentials
I received an MS degree at OSU in Mathemematics and a BS degree at OSU in Mathematical Science.

Awards and Honors
I graduated from OSU with a Bachelor of Science degree in mathematics with honors. Two years later, I graduated from OSU with a Master of Science degree in mathematics with honors. Between these two degrees, I have more hours in graduate courses that required to get them.

Past/Present Clients
I answered many questions at OSU, down in south seattle at a college, at a church in Corvallis, OR, as Safeway in Washington, and many other areas. I have answered over 8,500 right here, but only a little over 190 of them have been in differential equations.