Higher Order Linear ODEs¶

Much of the concepts from 2nd order linear ODEs will be applicable here in a more general and extended form. We consider all ODEs of the form

\[ \begin{aligned} y^{(n)} + p_{n-1}(x)y^{(n-1)} + p_{n-2}(x)y^{(n-2)} + \cdots + p_{1}(x)y^{(1)} + p_{0}y = r(x) \end{aligned} \]

With the coefficient of \(y^{(n)}\), this is called the standard form. The term on the right contains only functions of \(x\) (and/or constants).

An \(n^{th}\) order ODE that cannot be expressed in the above form will be called non-linear. If \(r(x) = 0\) throughout, the equation is called homogenous, and nonhomogenous otherwise.

Similar to second order case, for the homogenous equation, any linear combination of \(n\) independent solutions \(y_{1}, \ldots, y_{n}\) is also a solution (general solution), with the \(n\) solutions being linearly independent and forming a basis.

The Wronskian holds the same properties as the second order case, for the homogenous equations. However, its now defined as

\[ \begin{aligned} W = \detm{\begin{matrix} y_{1} &y_{2} &\cdots &y_{n}\newline y_{1}^{(1)} &y_{2}^{(1)} &\cdots &y_{n}^{(1)}\newline \vdots &\vdots &\cdots &\vdots\newline y_{1}^{(n-1)} &y_{2}^{(n-1)} &\cdots &y_{n}^{(n-1)}\newline \end{matrix}} \end{aligned} \]

\(n^{th}\) order linear homogenous ODE has no singular solutions.

Homogenous Linear ODE with constant coefficients¶

We consider the ODE

\[ \begin{aligned} y^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_{1}y^{(1)} + a_{0}y = 0 \end{aligned} \]

Assuming the solution as \(e^{\lambda x}\), we get the below characteristic equation after taking out the common term \(e^{lambda x}\)

\[ \begin{aligned} \lambda^{n} + a_{n-1}\lambda^{n-1} + \cdots + a_{1}\lambda + a_{0} = 0 \end{aligned} \]

The roots can be all real, real and repeated, some complex and some real, etc.

Real and distinct roots

Similar to the second order case, the general solution is a linear combination of \(n\) exponents

\[ \begin{aligned} y = c_{1}e^{\lambda_{1}x} + \cdots c_{n}e^{\lambda_{n}x} \end{aligned} \]

All complex roots

Complex roots must occur in pairs. Suppose one of such pairs is \(\lambda \pm i\omega\), then the part of general solution for just this pair will be

\[ \begin{aligned} y = c_{1}e^{\lambda x}\sin \omega x + c_{2}e^{\lambda x}\cos \omega x \end{aligned} \]

and similar terms will be added for all other pairs of complex roots.

Some distinct real and complex roots

This case is a combination of the above two cases. Hence the general solution will be a linear combination of few purely exponential terms, and few exponent and sinusoidal terms

\[ \begin{aligned} y = c_{1}e^{\lambda_{1}x} + \cdots + c_{k}e^{\lambda_{k}x}\sin \omega x + c_{k+1}e^{\lambda_{k}x}\cos \omega x \end{aligned} \]

Repeating real roots

Analogous to the second order case, we start multiplying all repeating roots by \(x\), \(x^{2}\) and so on

\[ \begin{aligned} y = c_{1}e^{\lambda_{1}x} + c_{2}xe^{\lambda_{1}x} + c_{3}x^{2}e^{\lambda_{1}x} + c_{4}e^{\lambda_{2}x} + c_{5}xe^{\lambda_{2}x} +\cdots \end{aligned} \]

In the above example, root $\lambda_{1}$ occurs thrice and we go upto $x^{2}$. Thus, if a root repeats $k$ times, we will have $k$ occurrences with the final occurrence containing $x^{k-1}$.

Repeating complex roots

Like repeating real roots, we start multiplying the terms by \(x\), \(x^{2}\) and so on

\[ \begin{aligned} y = c_{1}e^{\lambda_{1}x}\sin \omega_{1}x + c_{2}e^{\lambda_{1}x}\cos \omega_{1}x + c_{3}xe^{\lambda_{1}x}\sin \omega_{1}x + c_{4}xe^{\lambda_{1}x}\cos \omega_{1}x + \cdots \end{aligned} \]

In the above case, the complex root $\lambda_{1} \pm i\omega_{1}$ occurs twice.

Thus, depending on what the roots are, we can define the basis based on the above discussed rules.

Nonhomogenous Linear ODEs¶

All the methods discussed for solutions of second order nonhomogenous linear ODEs are applicable for higher orders as well. We will consider a linear ODE in its standard form

\[ \begin{aligned} y^{(n)} + p_{n-1}(x)y^{(n-1)} + p_{n-2}(x)y^{(n-2)} + \cdots + p_{1}(x)y^{(1)} + p_{0}y = r(x) \end{aligned} \]

Note that the coefficient of \(y^{(n)}\) is unity.

The general solution \(y = y_{h} + y_{p}\) where \(y_{h}\) is the solution to the corresponding homogenous equation, and \(y_{p}\) is a solution that satisfies the original nonhomogenous equation.

Method of Undetermined Coefficients¶

All rules followed are similar to those discussed in second order case. If the coefficients are constants, and \(r(x)\) is a continuous non zero function, the ODE is

\[ \begin{aligned} y^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_{1}y^{(1)} + a_{0}y = r(x) \end{aligned} \]

We follow the same three rules * Basic rule is same * Modification rule is slightly different. If a term of choice in \(y_{p}\) is already a solution of \(y_{h}\) (the corresponding homogenous equation solution), we multiply this term by \(x_{k}\) where \(k\) is the smalles positive integer such that this term times \(x^{k}\) is not a solution of the corresponding homogenous equation. * Sum Rule is same

Method of Variation of Parameters¶

We extend the formula discussed in the second order case

\[ \begin{aligned} y_{p}(x) = \sum_{k=1}^{n}y_{k}(x)\int \frac{W_{k}(x)}{W(x)}r(x)dx \end{aligned} \]

where \(W_{k}\) is obtained by replacing the \(k^{th}\) column of \(W\) with \((0, 0, \ldots, 0, 1)^{T}\) (i.e., the last row contains the entry 1).

The same formula can be verified for the second order case (where \(W_{1}\) and \(W_{2}\) simplified to a single term).