= Solution
For chosen positive terminal currents, define self-inductance by the coil's own flux linkage $\Lambda_j=L_jI_j$ and mutual inductance by $\Lambda_2=M I_1$ when only coil A carries current. With matching winding directions and terminal references,
$$
\boxed{L_A=\frac{\mu_0n_1^2A_t}{s},\qquad L_B=\frac{\mu_0n_2^2A_t}{s},\qquad M=\frac{\mu_0n_1n_2A_t}{s}.}
$$
The reciprocal flux-linkage relation gives the same $M$ from B to A, and $M^2=L_AL_B$ in this ideal fully coupled long-solenoid limit. The induced emf is $-d\Lambda/dt$; passive terminal voltage drops use the opposite sign, $v_A=L_A\dot I_A+M\dot I_B$ and similarly for B.
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