Basic facts are thos that can appear on a quiz and that must be committed to memory.
Order of a differential equation= largest number of times dependent variable is differentiated in one term.
The general solution of an nth order equation contains n constants to be determined from boundary conditions and thus needs at least n independent solutions.
Degree of a differential equation=highest power of dependent variable or its derivatives.
First degree equations are called Linear. Linear equations are important because their solutions obey the Principle of Superposition which states that if y1(x) and y2(x) solutions of the differential equation then any linear combination of them is also a solution.
A differential equation is Homogeneous in y if replacing y by αy does not change the equation. For linear equation it means that there is no term that is independent of y, no driving term.
A First Order equation that can be re-arranged into the form f(y)dy = g(x)dx is called a Separable equation. Its solution is found by integrating both sides.
A system executes Simple Harmonic Motion if it obeys the equation of motion
d2y + ω2y = 0
dt
or any equivalent. The motion of the system is then given by one of the equivalent forms
y(t)=Acos(ωt+φ)
y(t)=Bsin(ωt+ψ)
y(t)=Ccos(ωt)+Dsin(ωt)
where the constants are to be determined from the initial conditions.
Note that these forms are mathematically equivalent in the sense that it is easy to convert one to the other by simple trig. formulae.
If we write the solution in the standard form y(t)=Acos(ωt+φ) then A is the amplitude of the motion (the largest excursion from zero), ω is the angular frequency, and φ the phase of the motion.
The angular frequency is related to the regular frequency f and the period T by
ω=2πf=2π/T
Useful facts are a little more complex than basic facts. It is a good idea to know as many of them as you can but they will not normally appear on quizzes.
An first order linear differential equation, which can be written
dy+f(x)y=g(x)
dx
can be converted into a separable equation by multiplying both sides by the integrating factor e∫f(x)dx.
The equation then becomes
d (ye∫f(x)dx)=g(x)e∫f(x)dx
dx
which can then be integrated directly.