The solution in the Laplace Domain, with zero initial conditions, requires to solve the circuit simply replacing the inductors impedance effect \$v(t)=L\frac{di(t)}{dt}\$ with \$V(s)=sLI(s)\$

The circuits expressions are:

$$V1=R1I1+sL1(I1-I2)$$

$$sL1(I1-I2)=R2I2+sL2I2$$

$$V2=sL2I2$$

They can be simply rearranged as:

$$I1=(V1+sL1I2)/(R1+sL1)$$

$$I2=sL1I1/(R2+sL2+sL1)$$

From here, I1 and I2 can be obtained:

$$I2=V1sL1/((R1+sL1).(R2+sL2+sL1)-(sL1)2)$$

$$I1=V1/(R1+sL1)(1+(sL1)2/((R1+sL1).(R2+sL2+sL1)-(sL1)2))$$

And finally, the transfer function is calculated as:

$$V2/V1=1-R1I1/V1-R2I2/V1$$