2.1 Analytical solutions of first-order differential equations2.1.1 Differential equation solvable by simple integrals2.1.2 Homogeneous differential equations2.1.3 Inhomogeneous linear differential equations2.1.4 Nonlinear differential equations with separable variables2.2 Special functions and second-order differential equations2.2.1 Gamma function2.2.2 Hypergeometric functions2.2.3 Bessel differential equations2.2.4 Legendre differential equations and functions2.2.5 Airy functions2.3 Solutions of linear differential equations with constant coefficients2.3.1 Mathematical modeling of linear constant-coefficient differential equations2.3.2 Laplace transform-based solutions2.3.3 Solutions of inhomogeneous differential equations2.3.4 Solutions of differential equations with nonzero initial values2.4 Analytical solutions of ordinary differential equations2.4.1 Analytical solutions of simple differential equations2.4.2 Analytical solutions of high-order linear differential equations with constant coefficients2.4.3 Analytical solutions of linear time-varying differential equations2.4.4 Solutions of time-varying differential equation sets2.4.5 Solutions of boundary value problems2.5 Solutions of linear matrix differential equations2.5.1 Analytical solutions of linear state space equations2.5.2 Direct solutions of state space models2.5.3 Solution of Sylvester differential equation2.5.4 Kronecker product-based solutions of Sylvester differential equations2.6 Analytical solutions to special nonlinear differential equations2.6.1 Solvable nonlinear differential equations2.6.2 Nonlinear differential equations where analytical solutions are not available2.7 Exercises