Electronic – Differential equation from a Wheatstone bridge with Inductor

If R2 is K times R1 then C1 has to be K times C2. It's as simple as that unless I'm missing something obvious. It's that by inspection.

At any frequency the impedance ratio of R1 to XC1 must be equal to R2 to XC2.

\$\dfrac{R_1}{\frac{1}{sC_1}}\$ = \$\dfrac{R_2}{\frac{1}{sC_2}}\$

Therefore R1C1 = R2C2

Start with the Wheatstone bridge equation from wikipedia which is the subtraction of two voltage dividers.

$$ V_G = V_s( \frac{R_x}{R_3+R_x} - \frac{R_2}{R_1+R_2}) $$

Sensitivity is the derivative with respect to \$R_x\$. You'll notice immediately that the voltage divider that doesn't have the element of interest doesn't impact the sensitivity at all. The only function of the second voltage divider is to create a set point to compare the first voltage divider to. If we take the derivative with respect to \$ R_x\$ we get: $$ \frac{R_3}{(R_3+R_x)^2} $$ Note that I've chosen to ignore \$ V_s\$ as it is a scaling factor here. It should be noted that as your \$ V_s \$ increases, your sensitivity does increase as well. If you were to plot this you would see that there isn't really a normal value for the component under test that gives a maximum sensitivity because there is no inflection point in the graph.
Wolfram Alpha Plot of Sensitivity where \$R_3 =1 \$

This actually shows that the maximum sensitivity is where \$R_x\$ trends towards 0. This assumes you don't have negative value components.

If we plot the sensitivity equation with \$ R_3\$ as the variable and \$ R_x \$ as a constant, we'll find there is a maximum right at our constant value we used for \$R_x\$:
Wolfram Alpha Plot of Sensitivity where \$R_x=1\$

This is likely where the notion that you should keep \$R_3\$ as close to the same value as \$R_x\$ comes from. The other voltage divider again just needs to match closely with the test component's voltage divider to keep the voltage difference of 0 between them.

From these two plots we can conclude that if we want to design a Wheatstone bridge with the maximum sensitivity, we should have as small of component values as possible while still keeping all of the components at approximately the same value.