Solve the differential equation dx/dt = 6xlnx/t, Assume x,t>0, and use the initial condition x(1)=4.

Answer 1

Given:

dx/dt = ( 6x ln x ) / t

x(1) = 4

Find:

x(t) = ?

Solution:

dx / ( x ln x ) = 6 dt / t

u = ln x

x = e^u

dx = e^u du

e^u du / ( e^u u ) = 6 dt / t

du / u = 6 dt / t

ln u = 6 ln t

ln( ln x ) = 6 ln t + C

ln( ln 4 ) = 6 ln 1 + C

ln( ln 4 ) = C

ln( ln x ) = 6 ln t + ln( ln 4 )

ln( ln x ) = ln( t⁶ ln 4 )

ln x = t⁶ ln 4 = ln( 4^( t⁶ ) )

∴ x(t) = 4^( t⁶ )

I hope this helps! Message me in the comments with any questions, comments or concerns on the way I handled the problem above!

Answer 2

use the substitution u=te^t then use and integrating factor to solve. You will have to use E_i as part of the solution (exponential integral)