The value limx→∞g(x) is zero option (c) is correct.
What is a differential equation?
The differential equation involves the function and its derivative to represent a relationship between the functions and their derivative. In simple words, the differential equation is the mathematical relation that consists of one or more functions with its derivative.
We have a differential equation:
[tex]\rm \frac{dy}{dx} =y^2(4-y^2)[/tex]
[tex]\rm\frac{dy}{y^2(4-y^2)} = dx[/tex] (rearranging the equation to solve)
[tex]\int \rm\frac{dy}{y^2(4-y^2)} = \int dx[/tex]
[tex]\rm \frac{ln(|y+2|)}{16} -\frac{1}{4y} -\frac{ln(|y-2|)}{16} = x+c[/tex] (after integrating both sides)
For the given initial condition g(-2) = -1
When x = -2, y = -1
[tex]\rm \frac{ln(|-1+2|)}{16} -\frac{1}{4(-1)} -\frac{ln(|-1-2|)}{16} = -2+c[/tex]
[tex]\rm \frac{1}{4} -\frac{ln(|-3|)}{16}+2 = c[/tex]
The value of c = 2.18
The equation we get:
[tex]\rm \frac{ln(|y+2|)}{16} -\frac{1}{4y} -\frac{ln(|y-2|)}{16} = x+2.18[/tex]
As we can see the value of g(-2) is negative and from the graph of the above equation it goes to infinity and it touches the x-axis at the infinity.
The value [tex]\displaystyle \lim_{x \to \infty}g(x)[/tex] when x reaches infinity the value of function g(x) becomes zero. ie.
[tex]\displaystyle \lim_{x \to \infty}g(x) =0[/tex]
Thus, the value [tex]\displaystyle \lim_{x \to \infty}g(x)[/tex] is zero option (c) is correct.
Learn more about the differential equation here:
https://brainly.com/question/14620493
