~Courses in differential equations usually follow the calculus sequence of I, II, and III.

~In calculus I, you are required to solve only one type where the variables involved can be separated. This is called the "variables separable" method. (See link on integration formulas for examples)

~Solving all types of differential eqs is essential for most fields of engineering, specially mechanical and electrical. In all disciplines (including business related fields), whenever rates of change (derivatives) are involved, you will, most likely, encounter differential equations. (see example on Radiocarbon dating).

~Differential eqs are equations that involve derivatives and/or differentials.

~They occur in many areas.  Growth/decay, motion of masses, electrical circuit theory, acoustics, among many.

~First order diff. eqs. involve derivatives no higher than the first. These are categorized by types and require different techniques for each type.

~Second order diff. eqs. involve derivatives no higher than the second.
Along with the second derivatives, they could also involve the first derivatives. Techniques here are a little more sophisticated.

~Certain higher order diff. eqs. (with derivatives higher than two) are also covered. Also, systems of differential equations are studied. This requires much knowledge of matrices & how to work with them.

~More advanced types involve partial derivatives (covered in calculus III) and are necessary for the analysis of sound waves, water waves, and related acoustics (i.e., music).

~The techniques for solving partial differential eqs. are quite involved and much calculus background is necessary.

~One of the most interesting techniques for solving many of the basic types is by the use of Laplace Transformations. In simple terms, it transforms a calculus differential eq into an algebraic equation with no calculus. It is then solved in the algebra world then transformed back (using inverse Laplace Transformations) to the calculus world where the solution is given. (Many a calculus student would like that to happen to all of their calculus problems)

~Another interesting technique for the second order type is by the use of series (covered in calculus II). These are referred to as series solutions. The technique can be very long and involved but is a great application of series.

~In my opinion, one of the most interesting situations encountered in differential equations is the analogy between the simple LRC electrical circuits and the mechanical system counterparts. These occur in the study of the second order linear types with constant coefficients. Let me give you a brief description.

~The differential equation for the electrical system is
Ld²q/dt² +Rdq/dt +(1/c)q = E(t),
where q=charge (coulombs), c=capacitance (farads), dq/dt=current (amps), R=resistance (ohms), d²q/dt²=rate of change in current, L=inductance (henries) and E(t) is the total electromotive force of the circuit (volts).

~ In the mechanical system, we have an object moving through a medium and has a very similar type of differential equation looking like this:
Md²y/dt² + Rdy/dt +ky=F(t),
where d²y/dt²=acceleration, M=mass of the object (weight/g), dy/dt=velocily, R=resistance (proportional to the velocity), y=displacement, K=constant (proportional to the distance y traveled) (for a spring, it's Hooke's constant) and F(t) is total force acting on the system.

~One can therefore model a mechanical system using the similar electrical system and predict behavior quite nicely. Since the differential equations of both systems are quite alike, their solutions will be also. A very interesting analogy.

~A course in differential equations is usually not required for math majors (depending on the school). It certainly would be at MIT or Cal.Tech., since it is vital to most engineering programs.

Solving Basic differential equations by the variables separable method: In my opinion, the most useful use of the integration process is in solving basic differential equations where the variables of the equation can be separated. Having taught differential equations for a good number of years, I can tell you, first hand, that only a select few can be solved this way. Most others require specialize techniques that must be used. Courses in differential equations are extremely important to engineering in various fields. For our discussion, we will concentrate on solving acceleration, velocity, and distance problems using the method of variables separable. Let's take a look at an example.

EXAMPLE: A particle moves in a straight line with acceleration a=12t²+6t. It starts from rest with an initial velocity of -3 ft/sec (traveling in our designated negative direction). Find the equation for distance s at any time t.

Since acceleration measures the rate of change in velocity with time, it is a derviative, a=dv/dt. Also, velocity measures the rate of change in distance with time, it also is a derivative, v=ds/dt. These quantities need to be substituted into the equation to give the desired differential equation to solve. An equation which involves derivatives is called a differential equation.

Since a=dv/dt, we have the differential equation dv/dt = 12t²+6t. Separating the variables we get, dv = (12t²+6t)dt.(notice the separation of the variable on opposite sides of the equation)
Integrating both sides, ∫ dv = ∫ (12t²+6t)dt. Which gives us,
v = 12t³/3 + 6t²/2 + C. Since v = -3 when t=0, substituting these values in the equation for v determines C = -3.
Substitute -3 for C and we have, v = 4t³+3t²-3.
Since v = ds/dt, we have another diff. Eq., ds/dt=4t³+3t²-3.
Separating the variables & integrating, we get, s=t⁴+t³-3t+c.
Since the particle starts from rest, s=0 when t=0, so c=0.
Therefore, the equation for distance is, s = t⁴+t³-3t.

~NOTE: When separating the variables, make sure the differentials are "up stairs" and to the right of the expressions to be integrated.