Intergrating Factor method?

Question

Use the Intergrating factor method to solve the differential equation

dy/dx + y/x = x^2 + 3

Given that when x=1 y=1

Show that the equation x^3/2 +2x=5 has a root between the values 1 and 2.

use the Newton-Raphson method to find the root of this equation? to 3dp

Answer 1

dy/dx + y/x = x^2 + 3

I(x)=e^∫(1/x)dx

=e^(lnx)

=x

y=(1/x)∫x(x²+3)dx

=(1/x)((1/4)x^4+(3/2)x²)

=(1/4)x³+(3/2)x+C

Don't know about that second question.

EDIT: Last part was supposed to be:

y=(1/x)(∫x(x²+3)dx+C)

=(1/x)((1/4)x^4+(3/2)x²+C)

=(1/4)x³+(3/2)x+(C/x)