Proceed as in example 6 in section 2.3 to solve the given initial-value problem. dy dx + 2xy = f(x), y(0) = 6, where f(x) = x, 0 ≤ x < 1 0, x ≥ 1

The given initial value problem is

It is given that

y(0) = 6, where f(x) = x, 0 ≤ x < 1 0, x ≥ 1

Now the equation reduces to

 -------------(1)

Integrating factor=

Multiplying equation (1), both sides by integrating factor i.e

The equation reduces to

Integrating both sides, the equation reduces to

Now integrating the function which is on right hand side, we get

put and then differentiating,we get

2 x dx = dt

⇒x dx = dt/2

                                =

Replacing t by x², we get

   ---------------(2)

It is given that when x=0,y=6.

Putting these value in equation (2),we get

6=1/2 + C

⇒C= 6-1/2

⇒ C =11/2

The solution of the initial value problem is

step-by-step explanation:

the answer is; biodiversity resulting from few ancestors

adaptive radiation occurs when a population is subjected to a large natural environment with spatial ecological variations. the parts of the population in the different ecologies begin to adapt to their local environments. due to non-random mating across the large populations, this causes the different populations from the large population to speciate within their local environments. they, therefore, diverge to different species sharing a common ancestor.