A new exponential Chebyshev operational matrix of derivatives based on Chebyshev polynomials of second kind (ESC) is investigated. The new operational matrix of derivatives of the ESC functions is derived
and introduced for solving high-order linear ordinary differential equations with variable coefficients in
unbounded domain using the collocation method. As an application the introduced method is used to
evaluate Dawson's integral by solving its differential equation. The corresponding differential equation to
Dawson's integral is a boundary value problem with conditions tends to infinity. The obtained numerical
results are compared with the exact solution and showed good accuracy.