Ordinary Differential Equations 19 | Solution Space

For systems of linear differential equations, the solution set has a rigid structure: in the homogeneous case, all solutions form an n-dimensional vector space. That fact matters because it turns an infinite-dimensional-looking problem (functions of time) into a finite one—once n linearly independent solutions are known, every other solution is a linear combination of them.

The discussion starts with first-order linear systems written in the form X′(t)=A(t)X(t)+B(t), where A(t) is an n×n matrix and B(t) is an n-dimensional vector. Under the usual continuity assumptions on A(t) and B(t), solutions exist globally on the whole interval I (and, in the common global setting, on all of R). A special simplification appears for autonomous systems, where A(t) and B(t) are constant. In a concrete 2×2 example, the phase portrait reveals a fixed point at the origin and closed circular orbits around it. Those closed orbits correspond to periodic solutions, and the direction of motion (clockwise as time increases) is read directly from the vector field.

To handle the general (nonhomogeneous) case, the key move is to study the associated homogeneous system obtained by dropping the forcing term: X′(t)=A(t)X(t). The homogeneous structure brings linear-algebra tools to the foreground. If α(t) and β(t) are solutions, then α(t)+β(t) is also a solution because differentiation and matrix multiplication distribute over addition. Likewise, scaling a solution by any real number produces another solution. As a result, the set S0 of all continuously differentiable solutions to the homogeneous system forms a real vector space (indeed, a subspace of the function space).

The central theorem then pins down its dimension. For an n-dimensional linear system, S0 has dimension n. The reasoning uses the existence and uniqueness of solutions to initial value problems (via the Picard–Lindelöf theorem). Each initial condition X(t0)=x0 determines a unique solution curve in an “extended phase portrait” that includes time. To convert the dimension question into a linear-algebra statement, a linear map L is defined by projecting each solution α(t) to its value at a fixed time t0: L(α)=α(t0). Surjectivity follows because every x0 in R^n occurs as the value of some solution at t0. Injectivity follows from uniqueness: if two solutions agree at t0, they must coincide for all t in I. With L a linear isomorphism between S0 and R^n, the dimensions match, giving dim(S0)=dim(R^n)=n.

The takeaway is practical: for homogeneous linear ODE systems, finding n linearly independent solutions is enough to generate the entire solution space. The next step—set up for the following video—is to see how this plays out in an explicit example.