Differential equation of an OP AMP circuit with RC elements
As pointed out in the comments, the voltages at the op amp's inputs are zero and no current flows into them. So you have your current \$i(t)\$ going into the input node and flowing out of the output node. This zero voltage between the RC blocks allows you to separate the analysis into two circuits.

The input:
$$v_{in}(t) = R_1 i(t) + \frac{1}{C_1}\int i(\tau)d\tau $$
The output:
$$v_{out}(t) = -R_2 i(t) - \frac{1}{C_2}\int i(\tau)d\tau $$
Which gives a bit more information than what you wrote:
$$i(t)=\frac{v_{in}(t)-\frac{1}{C_1}\int i(\tau)d\tau}{R_1}$$
and
$$-\int i(\tau)d\tau = C_2\left(v_{out}(t)+R_2i(t)\right).$$
Then
$$i(t)=\frac{v_{in}(t)+\frac{C_2}{C_1}(v_{out}(t)+R_2i(t))}{R_1} $$
So
$$i(t)\left(1-\frac{C_2R_2}{C_1R_1}\right)=\frac{v_{in}(t)+\frac{C_2}{C_1}(v_{out}(t))}{R_1} $$
which allows you to write \$i(t)\$ as a function of \$v_{in}(t)\$ and \$v_{out}(t)\$.