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You get a crazy answer because you want to use numerical floats as constants, that Maxima's desolve tries to approximate by fractions. This is crazy : desolve will happily get you a "non-crazy"answer if you yse ymbolic constants, that you vcan substitute afterwards :

 sage: T=function("T")
sage: de=diff(T(x),x)==a+b*(T(x)-c)
sage: f=desolve(de,T(x),[0,c], ivar=x);f
(b*c + a*e^(b*x) - a)/b
sage: f.subs([a==1.54,b==-0.259,c==22])(x=7.098)

Close enough for government's work. BTW, you are aware that most decimals do not have an exact binary representation, sure ?