Theory

Systems from higher order equations

A system of ODEs looks like this:

$$\begin{array}{rl}{y}_1^{\prime }& =f_1(t,y_1,\dots ,y_n)\\ y_2^{\prime }& =f_2(t,y_1,\dots ,y_n)\\ ⋮& \\ y_n^{\prime }& =f_n(t,y_1,\dots ,y_n).\end{array}$$

The derivative of each variable $y_i$ is determined by its own function of the other variables, $f_i(t,y_1,\dots ,y_n)$.

One reason systems are important is that they offer an alternative to the language of order of a differential equation. Consider a general ODE of order $n$ given by the equation

$$F(t,y,y^{\prime },\dots ,y^{(n)})=0.$$

This ODE (and its complete families of solutions as well as IVPs) is equivalent to a first-order system in which higher derivatives are replaced with new variables:

$$\begin{array}{rl}F(t,y_0,y_1,\dots ,y_n)& =0\\ y_1& =y_0^{\prime }\\ y_2& =y_1^{\prime }\\ y_3& =y_2^{\prime }\\ ⋮& \\ y_n& =y_{n-1}^{\prime }.\end{array}$$

In particular, consider the $n^{\text{th}}$ order linear ODE:

$$y^{(n)}+a_{n-1}(t)y^{(n-1)}+\cdots +a_1(t)y^{\prime }+a_0(t)y=f(t).$$

By defining $y_0,\dots ,y_{n-1}$ with the formulas $y_0=y$ and $y_i=y^{(i)}$, we find an equivalent linear system:

$$\begin{array}{rl}{y}_0^{\prime }& =y_1\\ y_1^{\prime }& =y_2\\ ⋮& \\ y_{n-1}^{\prime }& =-a_{n-1}(t)y_{n-1}-\cdots -a_0(t)y_0+f(t).\end{array}$$

Notice that this system can be represented using matrix multiplication:

$${\left(\begin{array}{c}{y}_0\\ y_1\\ ⋮\\ y_{n-1}\end{array}\right)}^{\prime }=\left(\begin{array}{cccccc}0& 1& 0& 0& \dots & 0\\ 0& 0& 1& 0& \dots & 0\\ ⋮& & & ⋮& \dots & ⋮\\ -a_0(t)& -a_1(t)& -a_2(t)& -a_3(t)& \dots & -a_{n-1}(t)\end{array}\right)\left(\begin{array}{c}{y}_0\\ y_1\\ ⋮\\ y_{n-1}\end{array}\right)+\left(\begin{array}{c}0\\ 0\\ ⋮\\ f(t)\end{array}\right).$$

The last equation suggests vector-matrix formalism for systems of linear equations:

$${\mathbf{y}}^{\prime }(t)=\mathbf{A}(t)\mathbf{y}(t)+\mathbf{f}(t),\mathbf{A}(t)=\left(\begin{array}{ccc}{a}_{11}(t)& \dots & a_{1n}(t)\\ ⋮& & ⋮\\ a_{n1}(t)& \dots & a_{nn}(t)\end{array}\right),\mathbf{y}(t)=\left(\begin{array}{c}{y}_1(t)\\ ⋮\\ y_n(t)\end{array}\right),\mathbf{f}(t)=\left(\begin{array}{c}{f}_1(t)\\ ⋮\\ f_n(t)\end{array}\right).$$

This format is called the normal form for the linear system of ODEs.

Later sections, starting with homogeneous systems with constant coefficients, address the problem of extending our previous methods for second-order linear equations to the formalism for linear systems.

The methods we will extend:

• Homogeneous – complete family of solutions
• Inhomogeneous – variation of parameters for a particular solution
• Constant coefficients – real, repeated, and complex: exponential, linear factor, and sinusoidal solutions

Some new methods will be introduced to address the complexities of linear interaction between variables:

• Independence – to explain ‘complete’ family of solutions
• Eigenvectors – to describe proportionality between variables having the same overall exponential / sinusoidal behavior
• Matrix calculus – to write a general solution formula for constant coefficient systems

Homogeneous system

When the normal form of the system has nso $\mathbf{f}(t)$ term, it is called homogeneous:

$${\mathbf{y}}^{\prime }(t)=\mathbf{A}(t)\mathbf{y}(t),$$

or simply ${\mathbf{y}}^{\prime }=\mathbf{Ay}$.

The important thing about homogeneous systems is that the solutions form a vector space, meaning that solutions can be added and scaled while remaining valid solutions. For example, if ${\mathbf{y}}_1(t)$ and ${\mathbf{y}}_2(t)$ are vector solutions, then

$$C_1{\mathbf{y}}_1(t)+C_2{\mathbf{y}}_2(t)$$

is also a valid vector solution for any constants $C_1$ and $C_2$. Here’s a proof:

$$\begin{array}{rl}(C_1{\mathbf{y}}_1(t)+C_2{\mathbf{y}}_2(t){)}^{\prime }& =C_1{\mathbf{y}}_1^{\prime }(t)+C_2{\mathbf{y}}_2^{\prime }(t)\\ & =C_1\mathbf{A}(t){\mathbf{y}}_1(t)+C_2\mathbf{A}(t){\mathbf{y}}_2(t)\text{(separate solutions)}\\ & =\mathbf{A}(t)(C_1{\mathbf{y}}_1(t)+C_2{\mathbf{y}}_2(t))\text{(linearity of A).}\end{array}$$

The fact that solutions can be added to form valid solutions is called the superposition principle and it applies to linear homogeneous systems in the same way as it applies to $2^{\text{nd}}$ order linear equations.

Since the complete family of solutions of a system like ${\mathbf{y}}^{\prime }=\mathbf{Ay}$ forms a vector space, we would like to describe this family as the span of a specific set of independent solutions. Just as for any other vectors, a set of solutions ${\mathbf{y}}_1,\dots ,{\mathbf{y}}_n$ is an independent set when any equation

$$C_1{\mathbf{y}}_1+\cdots +C_n{\mathbf{y}}_n=0$$

can only be valid when $C_1=\cdots =C_n=0$.

A set of independent solutions is a complete set when every other solution vector can be written as a linear combination of them, i.e. it is in their span:

$$\mathbf{y}(t)={\alpha }_1{\mathbf{y}}_1+\cdots +{\alpha }_n{\mathbf{y}}_n,\text{some }{\alpha }_1,\dots ,{\alpha }_n.$$

In linear algebra, a complete set of independent vectors for a space is called a basis for the space. In ODE theory, a complete set of independent solution vectors is called a fundamental set of solutions. To reiterate: a fundamental set of solutions ${\mathbf{y}}_1,\dots ,{\mathbf{y}}_n$ is both independent and complete.

Solving ODE system

Solving an ODE system (finding the complete family of solutions) means: finding a fundamental set of solutions.

Given a fundamental set of solution vectors ${\mathbf{y}}_1,\dots ,{\mathbf{y}}_n$, we can arrange these vectors as the columns of a matrix called the fundamental matrix:

$$\mathbf{Y}(t)=\left(\begin{array}{ccc}{\mathbf{y}}_1& \cdots & {\mathbf{y}}_n\end{array}\right).$$

By linear algebra theory, these columns form a basis if and only if this matrix has nonzero determinant. The determinant of any fundamental matrix is called the Wronskian matrix $W(t)$.