SOLUTION: solve the initial value problem: please help with this differential equation {{{(dy)/(dx)}}}{{{""=""}}}{{{(e^(4x)+1)/(e^(4x)+4x+4)}}} (x>-1) y=1 when x=0 I have y= {{{(

 (x>-1) y=1 when x=0










Let u = e4x + 4x + 4
   du = (4e4x + 4)dx
   du = 4(e4x + 1)dx 
   dx =  

ln|u| + C

y = ln|e4x + 4x + 4| + C

And since x > -1 we can dispense with the absolute value:

y = ln(e4x + 4x + 4) + C 

Substitute the initial condition y=1 when x=0

        1 = ln(e4(0) + 4(0) + 4) + C
        1 = ln(1 + 0 + 4) + C 
        1 = ln(5) + C

1 - ln(5) = C

Substitute for C

y = ln(e4x + 4x + 4) + C

y = ln(e4x + 4x + 4) + 1 - ln(5)

You can leave it like that or 

Change the 1 to  so you can factor out 

y = ln(e4x + 4x + 4) +  - ln(5) 

y = [ln(e4x + 4x + 4) + 4 - ln(5)]

Edwin