Bäcklund transformations may be used also in ODE to solve non-linear problems; for instance, it's well known that for the equation
$$ \begin{align} \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega \tag{*} \end{align} $$$$ \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=\sin\omega \tag{*}\label{star} $$
we can define the transformation
$$ \begin{align} \frac{\mathrm{d}\omega}{\mathrm{d}t}-i\frac{\mathrm{d}\tau}{\mathrm{d}t}=2e^{i\lambda}\sin\left(\frac{\omega+i\tau}{2} \right) \end{align}, $$$$ \frac{\mathrm{d}\omega}{\mathrm{d}t}-i\frac{\mathrm{d}\tau}{\mathrm{d}t}=2e^{i\lambda}\sin\left(\frac{\omega+i\tau}{2} \right), $$
with $\tau(t)$ satisfying
$$ \begin{align} \frac{\mathrm{d}^2\tau}{\mathrm{d}t^2}=\sinh\tau, \end{align} $$$$ \frac{\mathrm{d}^2\tau}{\mathrm{d}t^2}=\sinh\tau, $$
and then (for any $\lambda$) with some choice of $\tau$ ($\tau=0$ in particular) we may find a solution for $\omega$. This works relatively easy because the equation (*)\eqref{star} is an autonomous equation, i.e., it does not contain the independent variable explicitly. On the other hand, if we consider
$$ \begin{align} \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=t\cdot\sin\omega \tag{**} \end{align} $$$$ \frac{\mathrm{d}^2\omega}{\mathrm{d}t^2}=t\cdot\sin\omega \tag{**}\label{starstar} $$
(or equations with any "forcing" $f(t)$ other that $f(t)=t$) the equation is no longer autonomous and it's not at all evident how to construct a Bäcklund transformation analogous to that of (*)\eqref{star}. (I'm assuming this method may help to solve the equation, which appears when considering some problems of deformations of elastic membranes.) The question is then how to construct a transformation that may help to solve (**)\eqref{starstar} analytically in terms of $\omega(t_0)$ and $\omega'(t_0)$, with $t_0$ some fixed real.
