Question #a16d8

1 Answer
Dec 19, 2016

The solution is y= x - 8ln|x| + 1

Explanation:

dy = (x- 8)/x dx

dy = (x- 8)x^-1dx

int(dy)= int(x - 8)x^-1dx

Integrate by parts on the right-hand side. Let dv= 1/x and u = x - 8. Then v = ln|x| and du = 1dx.

The integration by parts formula states that int(u dv) = uv - int(vdu).

Therefore:

int(x - 8)x^-1 = (x- 8)ln|x| - int(ln|x|)

We will now need to reintegrate using integration by parts.

We let u = ln|x| and dv = 1. du = 1/x dx and v = x.

int(ln|x|) = xln|x| - int(1/x xx x)

" "= xln|x| - x

" "=x(ln|x| - 1) + C

So, the complete integration, after integrating both sides, will be:

y = (x -8)ln|x| - (x(ln|x| -1)) + C

y= (x- 8)ln|x| - xln|x| + x + C

y = xln|x| - 8ln|x| - xln|x| + x + C

y= x - 8ln|x| + C

We want the equation that passes through (1, 2). Thus:

2 = (1- 8)ln|1| - 1(ln|1|) + 1 + C

2 = -7(0) - 1(0) + 1 + C

2 - 1 = C

C = 1

The solution to the differential equation is therefore y= x - 8ln|x| + 1

Hopefully this helps!