Respuesta :
The largest interval which includes x=0 for which a given initial-value problem has a unique solution is (-∞,3).
What is an equation?
An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
The given parameters are:
(x − 3)y'' + 4y = x,
y(0) = 0
y'(0) = 1
Divide the equation (x − 3)y'' + 4y = x through by (x - 3), therefore,
[tex]y''+\dfrac{4y}{x-3} = \dfrac{x}{x-3}[/tex]
Compare the above equation to the following equation,
y" + p(x) y' + q(x)y = g(x)
Then, you will get,
P(x)=4y/(x-3)
q(x)=0
g(x)=x/(x-3)
The domains of functions p(x) and g(x) are all sets of real values except 3.
This is represented as,
(-∞,3) ∪ (3,∞)
Using the unique existence theorem, we have:
The largest interval that contains x = 0 is (-∞,3).
Hence, the largest interval which includes x=0 for which a given initial-value problem has a unique solution is (-∞,3).
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