Solutions to the second-order equation
`m frac(d^(2) y)(d t^(2)) + b frac(dy)(dt) + ky =0`
typically model harmonic oscillators. The variables represent the following:
You may also see the form above written, differently, as:
`frac(d^(2) y)(d t^2) + p frac(dy)(dt) + qy =0`
because `m!=0`, in which case `p=b"/"m` and `k=k"/"m` are non-negative constants and a linear system representing this second-order equation is written as follows:
`frac( d bb(Y))(dt)= ((0, 1), (-p, -q))bb(Y)`
Our calculations convert the Second-Order Equation entered into a First-Order Linear System to generate the general solution found above, where we find the eigenvalues and eigenvectors. However, you can "guess a scalar solution" by noting that with an equation `a frac(d^(2) y)(d t^(2)) + b frac(dy)(dt) + cy =0` we can come up with the characteristic equation `as^2 + bs + c = 0` where the roots (eigenvalues) are instead found directly from the quadratic formula, where the general solution obtained is more formulaic. Blanchard, Devaney, and Hall say the following about this method, "Indeed this method is so efficient that one might be tempted to ask, 'Do we really need eigenvalues, eigenvectors, phase planes, and the rest of the ideas....?' The answer is 'no' provided we care only about formulas and not about a qualitative understanding of solutions. It is also important to remember that this method does not generalize to other linear systems" (329).
The Linearity Principle allows us to produce new solutions from known ones by adding solutions to each other and multiplying solutions by constants. This means that second-order equations of the form `a frac(d^(2) y)(d t^(2)) + b frac(dy)(dt) + cy =0` where `a`, `b`, `c` are arbitrary constants are said to be linear. Blanchard, Devaney, and Hall say, "More precisely these equations are homogeneous, constant-coefficient, linear, second-order equations." This is because of the constants `a`, `b`, `c` and the fact that the right side is equal to zero, making it homogeneous (325).
You may wish to review how this method fits into how we Classify Harmonic Oscillators, and how we find the General Solution in Scalar Form for Second-Order, Linear Homogeneous Equations.
Just as with the general solution to First-Order, Linear, Homogeneous Equations the computation of eigenvalues and eigenvectors is very important for finding the general solution in the case of Second-Order Linear, Homogeneous Equations because we are converting equations of this form into First-Order Linear Systems.
If you are unfamiliar with how to find eigenvalues and eigenvectors you may wish to visit the following vCalc pages:
In order to broaden your understanding we will provide an example:
Given the second-order equation
`frac( d^(2) y) (d t^(2)) + 7 frac(dy)(dt) + 10 y = 0`
we convert it into the first-order system
`frac( d bb(Y))(dt)= ((0, 1), (-p, -q))bb(Y) = ((0, 1), (-7, -10))bb(Y)`
where `-p =-7"/"1=-b"/"m` and `-q=-10"/"1=-k"/"m`.
This is the same thing as stating that `frac(dy)(dt) =v,` making `frac(dv)(dt)=-10y-7v.`Here, we see how the second-order equation is converted into two, separate equations with the substitution of a variable.
Solving for the characteristic polynomial yields
`lambda^2 + 7 lambda + 10 = 0`,
so `lambda_(1) =-2` and ` lambda_(2)=-5` with `vec(v)_1=((1),(-2))` and `vec(v)_2= ((1),(-5))` as the corresponding eigenvectors.
This leads to the general solution: `bb(Y) (t) = c_(1) e^(-2t) ((1),(-2)) + c_(2) e^(-5t) ((1),(-5))`
Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. 3rd ed. Belmont, CA: Thomson Brooks/Cole, 2006. Print.