solve the differential equation

Question

Consider the differential equation  with initial condition . First solve the differential equation to find the general solution , and then use it to evaluate .

Answer 1

xln(x)(dy/dx) = 1; (e,1)

dy/dx = 1/[xln(x)]

∫(dy/dx)dx = ∫dx/[xln(x)]

Let u=ln(x) so 1/u=1/ln(x), and let du=dx/x:

∫(dy/dx)dx = ∫du/u

y = ln(|u|)+C

y = ln(|ln(x)|)+C

Plugging in (e,1):

1 = ln(|ln(e)|)+C

1 = ln(|1|)+C

1 = C

Therefore, the solution to the differential equation is y=ln(|ln(x)|)+1

Plugging in x=e^e:

y(e^e) = ln(|ln(e^e)|)+1 = ln(|e|)+1 = 1+1 = 2, so y(e^e)=2

Hope this helped!