I Heavyside’s operational calculus vs. transforms

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Are there features of operational calculus (or operator methods) that are advantageous over transforms for DE? I know that the techniques are closely related.
 
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I’ve been poking around all day. I think I have convinced myself that there are situations where operation calculus would be useful.

I even found a great quote:
“Even Cambridge mathematicians deserve justice” -O. Heavyside
 

Thread 'A question about variables of integration'

I have the equation ##F^x=m\frac {d}{dt}(\gamma v^x)##, where ##\gamma## is the Lorentz factor, and ##x## is a superscript, not an exponent. In my textbook the solution is given as ##\frac {F^x}{m}t=\frac {v^x}{\sqrt {1-v^{x^2}/c^2}}##. What bothers me is, when I separate the variables I get ##\frac {F^x}{m}dt=d(\gamma v^x)##. Can I simply consider ##d(\gamma v^x)## the variable of integration without any further considerations? Can I simply make the substitution ##\gamma v^x = u## and then...
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