The present study is related to the design of a new mathematical model based on the Lane–Emden pantograph delay differential equation. The new model is obtained by using the sense of delay differential equation and standard Lane–Emden second-order equation. For the numerical solutions of the designed model, a well-known Bernoulli collocation method is implemented. In order to check the perfection and exactness of the designed model, three different nonlinear examples have been solved by using the Bernoulli collocation scheme. Furthermore, the comparison of the numerical results obtained by the Bernoulli collocation scheme with the exact solutions is also presented. Moreover, some numerical tables and the graphs of absolute error are plotted using different values of N for all problems.