Question

How do you find the primes denote derivatives with respect to x. Such as xy'+ 5y= 7x^2, y(2)= 5

Answer 1

Rewrite as

dy/dx + (5/x)*y = 7x

That is a first-order linear D.E. that can be solved using the method of integrating factors. In this case,
let P(x) = 5/x and Q(x) = 7x

Find the "integrating factor"
r(x) = exp(P dx) = 5 lnx

The solution is then
r(x)*y = integral of r*Q dx + C

Then use the y(2) = 5 information to compute C