If the owner of the railroad track and the county board or other public authority in interest fail to agree upon any of the matters or things mentioned in section 74-1337, either the owner or the county board or other public authority in interest, in the name of the county or other public authority in interest, may file an application with the Department of Transportation, setting forth such fact together with a statement of the change, alteration, relocation, or construction it wants, the estimated cost thereof, and such other facts as may be relevant and asking the department to enter an order directing that the change, alteration, relocation, or construction be made. The department shall proceed to hear the application in the manner provided by law, and if it finds that the application should be granted, it shall enter an order accordingly, designating in the order what portion of the expense of complying with the order shall be paid by the railroad carrier and what portion shall be paid by the county or other public authority in interest, if any.