Answer pleeeas

(3.1) … … …

Multiply the right side by x/x :

Substitute y(x) = x v(x), so that dy/dx = x dv/dx + v :

This DE is now separable. With some simplification, you get

Now you're ready to integrate both sides (on the left, the denominator makes for a smooth substitution), which gives

Solve for v, then for y (or leave the solution in implicit form):

(3.2) … … …

It may help to recognize this as a Bernoulli equation. Multiply both sides by :

Substitute , so that . Then you get a linear equation in z, which I write here in standard form:

Multiply both sides by an integrating factor, , which gives

and lets us condense the left side into the derivative of a product,

Integrate both sides:

Solve in terms of y :

(3.3) … … …

This DE is exact, since

are the same. Then the general solution is a function f(x, y) = C, such that

Integrating both sides of the first equation with respect to x gives

Differentiating this result with respect to y then gives

Then the general solution is

Given that y (1) = 4, we find

so that the particular solution is

hello i am doing good for you that it's summer in your country or state ; )

btw in my country we just have sun and rain as (because i live in the caribbean lol)

also, thx a lot for the points : )