Higher order Linear ODEs
Contents
Higher order Linear ODEs#
Wronskian#
We define a Wronskian of two functions as the determinant
Now consider the Wronskian of two solutions of a homogeneous ODE \(y'' + p(x)y' + q(x)y = 0\):
Differentiating it
In other words, the Wronskian satisfies the first-order linear ODE
whose solution is the Abel’s identity
In general, the Wronskian of a set of functions \(\{f_1(x), f_2(x), \dots, f_n(x) \}\) is defined by
Second-order linear ODE#
Reduction of Order - homogeneous#
Often we can find one solution of an ODE from inspection, which we can then use to find the other linearly independent solution.
And integrating
Can set C = 0 since we write the solution as a linear combination of u1 u2 anyway
Variation of parameters - inhomogeneous#
Let the general solution of the corresponding homogeneous equation be known
As we have done before, we use variation of parameters to set the ansatz for the particular solution:
where we have “promoted” the constants \(C_1, C_2\) to functions \(A(x), B(x)\). Differentiating it once:
We also require that
so eq. (158) becomes
Differentiating again
Now we substitute (161) and (162) into eq. (157):
We get
The solution of the system of equations (159) and (163) is given by
and after integrating we get
where the choice of the lower bounds of integration is irrelevant since it only changes the coefficients \(A\) and \(B\) (see below). Therefore, the solution of the inhomogeneous second-order ODE (157) is
General \(n\)-th order linear ODE#
We can directly extend the method of variation of parameters we used to solve second-order linear ODE to \(n\)-th order ODEs, which are of the form
Let \(\{y_1, y_2, \dots, y_n\}\) be solutions of the homogeneous ODE, i.e. \(\mathcal{L}[y_i] = 0\). We seek solutions of the inhomogeneous ODE in the form of ansatz
where \(C_i(x)\) are functions that we need to find. As before, we differentiate and define conditions:
These \(n\) equations define a system of linear equations with unknowns \(C_1, \dots, C_n\):
A sufficient condition for the existence of a solution of the system of equations (130) is that the determinant of coefficients is nonzero for each value of \(x\). This is equal to the Wronskian \(W[y_1, \dots, y_n]\) and we know it is nonzero for all \(x\) because the set \(\{y_1, \dots, y_n \}\) is linearly independent. We determine \(C'_1, \dots, C'_n\) using Cramer’s rule:
where \(W_i\) is the determinant obtained from \(W\) by replacing \(i\)-th column with the RHS column \((0, 0, \dots, r)^T\). We integrate this and substitute the result in eq. (129) to get the particular solution of (188)
where \(x_0\) is arbitrary.
Linear ODEs with Constant Coefficients#
In this section we focus on second-order ODEs where functions \(p(x), q(x)\) and \(r(x)\) are now constants. We therefore write Eq. inhmg2ndode
as
Homogeneous case - method of characteristic polynomial#
We use the method of characteristic polynomial which can be used for any such ODE of any order. A homogeneous linear ODE of first order with constant coefficient \(a_0\)
is separable and its integral is \(y(x) = Ce^{\lambda x}\), where \(\lambda\) is an unknown constant. We find it by plugging \(y\) back into the ODE:
This is the characteristic equation of the ODE. We find \(\lambda = -a_0\).
Consider now a second-order ODE with constant coefficients \(a_1, a_0\)
Inspired by the solution of the first-order equation, we assume the solution of the form \(y(x) = Ce^{\lambda x}\). We plug it back into the ODE and get the characteristic equation
The roots of this quadratic equation are
so we have three cases depending whether \(D > 0, D = 0\) or \(D < 0\), keeping in mind that we need linearly independent solutions to form the basis:
\( \lambda_{1,2} \in \mathbb{R}, \lambda_1 \neq \lambda_2: \quad y(x) = C_1 e^{\lambda_1 x} + C_2e^{\lambda_2 x} \)
\( \lambda_{1,2} \in \mathbb{R}, \lambda_1 = \lambda_2: \quad y(x) = (C_1 + C_2 x)e^{\lambda_1 x} \)
\( \lambda_{1,2} \in \mathbb{C}, \lambda_2 = \lambda_1^*: \quad y(x) = C_1 e^{\lambda_1 x} + C_2e^{\lambda_1^* x} \)
Hint: Diagonalisation
The method of characteristic polynomial might remind us of the process of finding eigenvalues. Indeed, we can think of finding solutions in the form of an exponential function as a type of diagonalisation.
The exponential function can be thought of as an eigenvector of the derivative operator and \(\lambda\) as the eigenvalue. If \(\lambda_1, \lambda_2\) are roots of the characteristic equation, we can write
Let us now generalise this to an \(n\)-th order ODE. The characteristic equation of eq. (131) is again obtained by substituting \(y = e^{\lambda x}\):
For any distinct real root \(\lambda\), one solution is \(y = e^{\lambda x} \)
For any complex root \(\lambda = \gamma + i \omega\), its conjugate \(\lambda^* = \gamma - i \omega\) is also a root:
Multiple real roots: if a real root is repeated \(m\) times, the \(m\) corresponding linearly independent solutions are
Multiple complex roots: if a complex root \(\lambda = \gamma + i \omega\) is repeated, the corresponding linearly independent solutions are