Systems of Differential Equations

Systems of differential equations can be solved using linear algebra, specifically the calculation of eigenpairs.

Formulation

Consider a system of differential equations like:

y'₁ = ay₁ + by₂

y'₂ = cy₁ - dy₂

Reformulate the system as:

Now consider a higher order differential equation like x''' + ax'' - bx' + cx = 0.

Define a set of substitutions:

• y₁ = x, so y'₁ = x'.

• y₂ = x', so y'₂ = x''.

• y₃ = x'', so y'₃ = x'''.

The original equation is now y'₃ + ay₃ - by₂ + cy₁ = 0, or equivalently y'₃ = -ay₃ + by₂ - cy₁. Use this along with the substitution definitions for y'₁ and y'₂ to create the following reformulation:

Solution

The general solution to a single differential equation like y' = ry is y = ce^rt where c is some constant.

The general solution to a system of differential equations like the above is of the form:

where each r_n is unique and each c_n is a vector matched to a specific r_n. Furthermore, each c_n is orthogonal to the others. The k_n are simply constants reflecting all linear combinations of these independent components.

These pairs of r_n and c_n are eigenpairs.

Repeated Eigenvalues

If there are repeated eigenvalues, it must be assumed that there is an additional independent solution. Consider a system with two variables. The repeated eigenpair will be notated r and c rather than r₁ and c₁. The general solution will have the form:

y = k₁e^rtc + k₂(te^rtc + e^rtd)

where d is another, independent vector that satisfies the equation:

(A - rI)d = c

This is rather like solving for the eigenvector in the first place. Rather than identifying a linear combination that maps (A - rI) to the zero vector, d maps it to eigenvector itself.

Complex Eigenpairs

A complex eigenpair can be refactored using Euler's equation (i.e., e^iθ = cosθ + isinθ).

Let the eigenvalue be a + bi; the eigenvector will be left as c. The solution then takes the form y = e^{(a + bi)t}c, but this can be refactored as y = e^at e^bit c and finally as y = e^at (cos(bt) + isin(bt)) c.

By multiplying out the trigonometric functions and c, then factoring out the imaginary components, the solution becomes:

The general solution however is formed by combinations of the independent components. The second component is used as a real vector, rather than an imaginary one.

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