Choose an application of engineering where differential equations are used.

RUNNING HEAD: DIFFERENTIAL EQUATIONS

DIFFERENTIAL EQUATIONS 3

Homogeneous linear equations

An equation is referred to as homogeneous if the right side is 0 and its linear because the equation is a sum of the available derivatives. A homogeneous linear equation has two solution and also arbitrary constants. Homogeneous linear equations therefore have “zero” solutions to the given set of variables and are solved by Gaussian elimination.

An example of a linear homogeneous equation is:

Example; Solve

The characteristic equation is; solving the characteristic equation

Gives, so

The general solution is

Getting the derivatives

Thus,

Importance of homogeneous linear equations in engineering

Used in a harmonic oscillator in classical mechanics where when it is displaced from its equilibrium position, a restoring force is experienced which is directly proportional to the displacement. A homogeneous second order linear differential equation is used in the equation of motion an ideal spring with a spring constant is defined by the simple harmonic oscillation with a fundamental equation shown below:

Homogeneous differential equations are applied in the charging and discharging of an RC circuit. RC circuits consist of capacitors and resistors connected to a voltage source in which the capacitor is charged exponentially with time. At full charge, the current stops flowing. This process can be described using differential equations as shown below. The equation for voltage against time when a capacitor is being charged through a resistor is:

The discharging process is characterized by a reducing voltage hence a decreasing rate of discharge. The voltage on the capacitor being discharged through a resistor is given by:

(Croft, Davison & Hargreaves, 2001).

References

Croft, T., Davison, R., & Hargreaves, M. (2001).Engineering mathematics: A foundation for electronic, electrical, communications, and systems engineers. Harlow, England: Prentice Hall.

Kelly, S. G. (2009). Advanced engineering mathematics with modeling applications. Boca Raton: CRC Press.

Singh, K. (2003). Engineering mathematics through applications. Basingstoke: Palgrave Macmillan.

Stroud, K. A., & Booth, D. J. (2003).Advanced engineering mathematics. Basingstoke: Palgrave Macmillan.

Xie, W.-C.(2010). Differential equations for engineers. Cambridge: Cambridge University Press.