In this work, we examine cognitive radio networks, where secondary users may act as relays for messages sent by the primary user, hence offering performance improvement of primary transmissions, while at the same time obtaining more transmission opportunities for their own data. In particular, assuming the broadcast packet erasure model with feedback, we investigate the capacity of the fundamental cooperative cognitive radio network which consists of one primary and one secondary transmitter-receiver pairs. The primary transmitter is the owner of the channel and as such, we intend to keep its operations simple and to avoid increasing its storage requirements. Specifically, the primary transmitter does not receive data sent by the secondary transmitter and does not perform any coding operations. On the other hand, the secondary transmitter can overhear primary transmissions and is allowed to perform any coding operations. We develop an outer bound to the capacity of the fundamental cooperative cognitive radio network under consideration. Then, we propose a coding-scheduling algorithm suitable for this type of networks, which involves only XOR network coding operations. The complexity of the scheduling decisions of the proposed algorithm depends on the channel statistical parameters and three cases, depending on the relations between channel erasure probabilities, are distinguished. For the first two cases the rate region of the proposed algorithm coincides with the developed capacity outer bound, hence the algorithm is capacity achieving. For the third case, the rate region of the proposed algorithm is not identical to the outer bound; however, numerical results show that it is fairly close to the derived outer bound for a wide range of the statistical parameters of the system.