Applications to differential equations

Application to Differential Equations

Linear algebra provides essential tools for solving systems of linear differential equations. Since differentiation is a linear operator, it can be represented as a matrix acting on a function space.

Systems of First-Order Equations

A system of n linear first-order differential equations can be written in matrix form:

x′(t) = Ax(t)

where x(t) is a vector of unknown functions and A is an n × n constant coefficient matrix.

Solution Method

If A has n linearly independent eigenvectors v₁, ..., vₙ with corresponding eigenvalues λ₁, ..., λₙ, the general solution is:

x(t) = c₁e^(λ₁t)v₁ + c₂e^(λ₂t)v₂ + ... + cₙe^(λₙt)vₙ

where c₁, ..., cₙ are constants determined by initial conditions.

Example

For the system:

x₁′ = 3x₁ + x₂
x₂′ = x₁ + 3x₂

The coefficient matrix A = [[3,1],[1,3]] has eigenvalues λ₁ = 4, λ₂ = 2 with eigenvectors v₁ = [1,1]ᵀ and v₂ = [1,−1]ᵀ. The solution is:

x(t) = c₁e^(4t)[1,1]ᵀ + c₂e^(2t)[1,−1]ᵀ

Higher-Order Equations

An nth-order linear differential equation can be converted to a system of n first-order equations, making eigenvalue methods applicable. This is fundamental in engineering, physics, and control theory.