In the same way you can to find a solution to your 1st, more general equation (with appropriate abbreviations):

r = (a0 + 
Integrate[ u[t]/f[t]  Exp[Integrate[(A + D[f[tt], tt])/f[tt], {tt, c2, t}]], {t, c1, x}])
Exp[-Integrate[(D[f[tt], tt] + A)/f[tt], {tt, c2, x}]]

In[57]:= f[x] D[r, x] + r (D[f[x], x] + A) // FullSimplify

Out[57]= u[x]

Hi Ewelina,

in the attached notebook I outline a step by step solution of your equation , but using lots of abbreviations.

Note the introduction of constants of integration c1, c2 and a0. These must be fixed by your initial-conditions.

I did NOT look at the resulting integrals, whether they could be solved and whether they yield the solutions given by DSolve (they should do), I leave this up to you.

Attachments:

wcomUntitled-3.nb

WolframAlpha doesn't appear to support user named functions. Uppercase variable names sometimes also cause problems. There is also a line length limit that sometimes requires a bit of skill in dividing a problem into parts or by solving problems one stage at a time. It does support a subset of Mathematica notation and at times this has let me solve problems that I couldn't otherwise find a way to explain to WolframAlpha

DSolve[(a1+(a2+a3)E^(-2x))?'[x]+?[x](D[a1+(a2+a3)E^(-2x),x]+? g/?)== -(? m^2/?)b v*E^-x ,?,x]

or

WA link

returns a complicated exact solution. You might be able to see something in that or might be able to simplify that.