Once you understand the basic idea of the method you can apply it to differential equations of any order.
First Order Case
First consider the first order differential equation,
$$ a(t)\frac{d y}{dt}+b(t) y= F(t) ,$$
and assume that $y_{\mathrm h}$ solves the homogeneous case (i.e. $F(t)=0$).
We can write any function in the form of the product,
$$ \boxed{f(t) = A(t) \cdot y_{\mathrm h}(t)},$$
if we substitute this product into the differential equation we will be able to exploit the fact that $y_{\mathrm h}$ solves the homogeneous equation to make certain simplifications.
$$a(t) \frac{d}{dt} \left( A(t) y_{\mathrm h}(t) \right) + b(t) \left( A(t) y_{\mathrm h}(t) \right) = F(t),$$
$$\Rightarrow a(t) \left( \frac{dA}{dt} y_{\mathrm h}(t) + \color{blue}{A(t) \frac{d y_{\mathrm h}}{dt}}
\right) + \color{blue}{b(t) A(t) y_{\mathrm h}(t) } = F(t),$$
$$\Rightarrow a(t) \frac{dA}{dt} y_{\mathrm h}(t) + A(t) \color{blue}{\left[ a(t) \frac{d y_h}{dt}+b(t) y_{\mathrm h}(t) \right]} = F(t),$$
the blue expression is equal to $\color{blue}{0}$ by the definition of $y_{\mathrm{h}}$ leaving us with,
$$\boxed{a(t) \frac{dA}{dt} y_{\mathrm h}(t) = F(t)},$$
If we can solve this equation for $A$ then we will have found a particular solution, in the form of $f$, to the original differential equation.
Example for 1st order case
As an example of the preceding section, consider the differential equation,
$$ \frac{dy}{dt}+y = \sin(t).$$
The homogeneous solution is $ y_{\mathrm{h}}(t) = y_0 \exp(-t)$.
The differential equation for $A$ is then,
$$ \frac{dA}{dt} = e^t \sin(t),$$
so that we have,
$$\boxed{ A(t) = \int^t e^{t'} \sin(t') dt'} ,$$
and the particular solution to the original equation is given by,
$$ \boxed{f(t) = f_0 e^{-t} \int^t_0 e^{t'} \sin(t') dt'}. $$
Second Order Case
In the second order case we are solving,
$$ a(t) y'' + b(t) y' + c(t) y = F(t),$$
which generally has two linearly independent solutions ( $y^{(1)}_{\mathrm{h}}$ and $y^{(2)}_{\mathrm{h}}$).
We can then write any function as,
$$ f(t) = A(t) y^{(1)}_h + B(t) y^{(2)}_h(t),$$
substituting this in the differential equation will gives us differential equations for $A$ and $B$ just like in the first order case. For the third order case we write $f(t)$ as a linear combination of three linearly independent homogeneous solutions and the pattern continues for higher order differential equations.
This answer isn't yet complete I'll be coming back later to fill in some details