differential equation

Question

consider the differential equation x*ln(x) dy/dx=1 with initial condition (e,1). First solve the differential equation to find the general solution y(x), and then use it to evaluate y(e^e).

Answer 1

xln(x) dy/dx = 1

xln(x) dy = dx

dy = dx/(xlnx)

∫dy = ∫1/(xln(x)) dx

y = ln(ln(x)) + C

Given (e,1) then:

1 = ln(ln(e)) + C

1 = ln(1) + C

1 = 0 + C therefore C = 1

So y = ln(ln(x)) + 1

y(e^e) = ln(ln(e^e)) + 1 = ln(e•ln(e)) + 1 = ln(e) + ln(ln(e)) + 1 = 1 + ln(1) + 1 = 1 + 0 + 1 = 2