It shall be the responsibility of the state and the appropriate departments and agencies thereof to discover methods and procedures by which the mental and/or physical health of the child in custody may be improved and, with the consent of the co-custodians, to apply those methods and procedures. The co-custodians other than the state shall have no financial responsibility for the child committed to their co-custody except as they may in written agreement with the state accept such responsibility. At any time after the commitment of such child they may inquire into his or her well-being, and the state and any of its agencies may do nothing with respect to the child that would in any way affect his or her mental or physical health without the consent of the co-custodians. The legal status of the child may not be changed without the consent of the co-custodians. If it appears to the state as co-custodian of a child that the health and/or welfare of such child is impaired or jeopardized by the failure of the co-custodians other than the state to consent to the application of certain methods and procedures with respect to such child, the state through its proper department or agency may petition the court for an order to proceed with such methods and procedures. Upon the filing of such petition a hearing shall be held in open court, and if the court finds that such petition should be granted it shall issue the order.