Show that if a₁₁, a₁₂, a₂₁, and a₂₂ are constants with a₁₂ and a₂₁ not both zero, and if the functions g₁ and g₂ are differentiable, then the initial value problem
x¹ = a₁₁x₁ + a₁₂x₂ + g₁(t), x₁(0) = x₀₁, x² = a₂₁x₁ + a₂₂x₂ + g₂(t), x₂(0) = x₀₂
can be transformed into an initial value problem for a single second-order equation. Can the same procedure be carried out if a₁₁, …, a₂₂ are functions of t?