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

answer: -1

step-by-step explanation:

if x = 1, then the answer is 3.5