Contour integral solutions of linear differential equations which include a generalization of the Airy integral

The Airy integral is a well-known contour integral solution of Airy’s equation which has several applications and which has been used for mathematical illustrations due to its interesting properties. We present and derive properties of two families of contour integral solutions of linear differential equations, where one family includes the Airy integral and Airy’s equation, such that the family generalizes known properties of the Airy integral which include exponential decay growth in a certain sector. The second family includes a known example and contains a subfamily with interesting properties where a separate analysis of three pairwise linearly independent contour integral solutions of a particular equation is given.