Calculus II modeling with differential equations

Question

Verify that every member of the family of functions y=(lnx+C)/x  is a solution of the differential equation x²y' + xy = 1

a) Find a solution of the differential equation that satisfies the initial condition 𝑦(7)=6. y=

b) Find a solution of the differential equation that satisfies the initial condition 𝑦(6)=7. y=

Answer 1

 y=(lnx+C)/x 

y'=[(1/x)*x-(lnx+C)]/x²=(1-lnx-C)/x²

Now plug in y and y' into equation x²y' + xy = 1

x²((1-lnx-C)/x² ) + x( lnx+C)/x = 1-lnx-C-lnx+C=1

So  y=(lnx+C)/x  is a general solution of the equation x²y' + xy = 1

Now

a) y(7) =6 and we find C.

6=(ln7+C)/7

42=ln7+C

C=42-ln7

So the particular solution with condition y(7)=6 is

y=(lnx+42-ln7)/x = (ln(x/7)+42)/x

b) is similar