In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are usi...In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.展开更多
Keats's poems are famous for beauty,and Ode to Autumn is generally regarded as one of his most beautiful poems.In this ode,Keats presents a vivid picture of the harvesting autumn,in which many forms of beauty can ...Keats's poems are famous for beauty,and Ode to Autumn is generally regarded as one of his most beautiful poems.In this ode,Keats presents a vivid picture of the harvesting autumn,in which many forms of beauty can be captured.The present paper attempts to conduct an analysis of the beauty in Keats's Ode to Autumn,hopefully it could help readers better understand Keats and his works.展开更多
文摘In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.
文摘Keats's poems are famous for beauty,and Ode to Autumn is generally regarded as one of his most beautiful poems.In this ode,Keats presents a vivid picture of the harvesting autumn,in which many forms of beauty can be captured.The present paper attempts to conduct an analysis of the beauty in Keats's Ode to Autumn,hopefully it could help readers better understand Keats and his works.
