According to Wikipedia, "For linear evolution equations without spatial dependency, such as a harmonic oscillator, Duhamel's principle reduces to the method of variation of parameters technique for solving linear inhomogeneous ordinary differential equations." However, I don't feel a very strong resemblance.
In order to solve such an ODE by variation of parameters, i.e., $y'' + \frac{1}{x}y' - \frac{1}{x^2}y = 3$, I would...
- Find the homogeneous solution: In the above example $y_h = \frac{A}{x} + Bx$
- Change the integration constants to functions of $x$: $y = \frac{v}{x} + wx$
- Take a derivative: $y' = \frac{v'}{x} + xw' - \frac{v}{x^2} + w$
- Choose a constraint to plug into derivative: $\frac{v'}{x} + xw' = \frac{3x}{2} \implies \tilde y' = \frac{3x}{2} - \frac{v}{x^2} + w$
- Take another derivative: $\tilde y'' = \frac{3}{2} + \frac{2v}{x^3} - \frac{v'}{x^2} + w'$
- Solve system of ODEs consisting of the constraint and the original ODE with $y$, $\tilde y'$, and $\tilde y''$ plugged in: $v' = \frac{3x^2}{4}, w' = \frac{3}{4}$
- Integrate $v$ and $w$, and plug back into guess: $y = \frac{x^3 + Bx^2 + A}{x}$
All of this is just to find the general solution, and then initial conditions can be applied to further refine the solution.
In order to solve the ODE by Duhamel's principle, the process would be completely different. It would involve finding the homogeneous solution, as do many solution processes, but I would need to include initial conditions in the problem statement to make it work. Duhamel's principle would also not involve introducing a constraint into the problem, nor would it require solving a system of ODEs. I would simply move the inhomogeneous term over to the initial condition, changing $x$ to a dummy variable in the process, and integrate the solution over that dummy variable from $0$ to $x$.
How are these two seemingly disparate solution processes actually identical in the special case where they both apply?