Question

PLZ HELP W SOME AP CALC QUESTIONS WILL MARK BRAINLIEST

Answer 1

1) The complete table is shown in the attachment .

2) The direction field is shown in attachment.

3) The given differential equation is:

$\frac{dy}{dx} = \frac{xy}{3}$

We separate variable to get:

$\frac{dy}{y} = \frac{1}{3} xdx$

We integrate both sides wrt x to get:

$\int \frac{1}{y} dy =\frac{1}{3} \int xdx$

This implies that:

$ln(y) = \frac{ {x}^{2} }{6} + ln(c)$

$y =k {e}^{ \frac{ {x}^{2} }{6} }$

Or

$f(x) =k {e}^{ \frac{ {x}^{2} }{6} }$

We apply the initial condition:

$f(0) = 4$

$4 =k {e}^{ \frac{ {0}^{2} }{6} }$

$4 = k$

The particular solution is:

$y =4 {e}^{ \frac{ {x}^{2} }{6} }$

4) See attachment