As we show in this paper, the identity for the Gaussian curvature of a surface associated with a

first-order ordinary differential equation (ODE) in the first-order jet bundle is described by an inhomogeneous Burgers' equation:

Consider the

first-order ordinary differential equationIn their scheme, the fractional differential equation can be converted into a set of

first-order ordinary differential equations which can be solved by the Runge-Kutta (RK) method or the trapezoidal integration rule, and so forth.

We are interested in initial value problems related to systems of

first-order ordinary differential equations (ODEs) of the form

Using step-by-step examples he covers differential equations,

first-order ordinary differential equations, linear second-order and systems of first-order differential equations, Sturm-Liouville problems, Fourier series and integrals, partial differential equations, applications of partial differential equations in chemical engineering, dimensional analysis and scaling of boundary value problems, selected numerical methods and available software packages.

In this system, for underlying Lie algebra with peculiar realization, we obtain seven linearly independent invariants whose invariant representation is in the form of system of

first-order ordinary differential equations. These invariants contain four first-order differential invariants and three second-order differential invariants.