General solution of linear system

The general solution of a linear system ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}$ is a formula that captures every solution as a linear combination of a fixed set of building-block solutions. For an $n\times n$ system, you need $n$ linearly independent solutions ${\mathbf{x}}_1(t),\dots ,{\mathbf{x}}_n(t)$, called a fundamental set, and the general solution is

$\mathbf{x}(t)=c_1{\mathbf{x}}_1(t)+c_2{\mathbf{x}}_2(t)+\cdots +c_n{\mathbf{x}}_n(t)$

with $c_1,\dots ,c_n$ arbitrary constants. Pick the constants to satisfy initial conditions.

This is the vector analog of $y=c_1{y}_1+c_2{y}_2$ for a second-order scalar ODE — same structure, more dimensions.

Why $n$ solutions exactly

The set of all solutions of ${\mathbf{x}}^{\prime }=A(t)\mathbf{x}$ is a vector space (you can add solutions and scale them — both still solve the system, by linearity). The dimension of that vector space is exactly $n$, the number of equations. So a basis of $n$ linearly independent solutions spans the whole space, which is what “general solution” means.

The dimension being $n$ is a consequence of the existence and uniqueness theorem: the initial state $\mathbf{x}(t_0)\in {\mathbb{R}}^n$ has $n$ free entries, so the solution space has $n$ degrees of freedom.

Linear independence — the Wronskian

To verify that $n$ candidate solutions ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_n$ are independent, form the Wronskian matrix by stacking them as columns:

$W(t)=\left[\begin{array}{cccc}{\mathbf{x}}_1(t)& {\mathbf{x}}_2(t)& \cdots & {\mathbf{x}}_n(t)\end{array}\right]$

The solutions are linearly independent if and only if $\det W(t_0)\ne 0$ at any single point $t_0$. (Once nonzero somewhere, it stays nonzero everywhere — Abel’s theorem.) See Linear independence of vector functions for the careful version.

Fundamental matrix

When you collect a fundamental set as columns of a matrix $\Phi (t)=W(t)$, that matrix is called a fundamental matrix for the system. The general solution is then

$\mathbf{x}(t)=\Phi (t)\mathbf{c},\mathbf{c}=\left[\begin{array}{c}{c}_1\\ ⋮\\ c_n\end{array}\right]$

To solve the IVP $\mathbf{x}(t_0)={\mathbf{x}}_0$, just solve $\Phi (t_0)\mathbf{c}={\mathbf{x}}_0$ for the constant vector — a single $n\times n$ linear system.

Constant-coefficient case

For ${\mathbf{x}}^{\prime }=A\mathbf{x}$ with constant $A$, the building blocks come from the eigenvalues:

• Each real eigenvalue $\lambda$ with eigenvector $\mathbf{v}$ contributes one solution $\mathbf{x}(t)=\mathbf{v}{e}^{\lambda t}$.
• A complex conjugate pair $\alpha \pm i\beta$ contributes two real solutions $e^{\alpha t}(\mathbf{u}\cos \beta t-\mathbf{w}\sin \beta t)$ and $e^{\alpha t}(\mathbf{u}\sin \beta t+\mathbf{w}\cos \beta t)$, where $\mathbf{u}+i\mathbf{w}$ is the complex eigenvector.
• A repeated eigenvalue with deficient eigenspace contributes solutions involving $t{e}^{\lambda t}$, $t^2{e}^{\lambda t}$, etc., built from generalised eigenvectors.

See Distinct real eigenvalues case, Complex conjugate eigenvalues case, and Repeated eigenvalues case for the detailed constructions. In every situation, the goal is the same: produce $n$ linearly independent solutions and stack them.

Worked example

${\mathbf{x}}^{\prime }=\left[\begin{array}{cc}1& 1\\ 4& 1\end{array}\right]\mathbf{x}$.

Characteristic polynomial: $(1-\lambda {)}^2-4=0$, so ${\lambda }^2-2\lambda -3=0$, giving ${\lambda }_1=3$ and ${\lambda }_2=-1$.

Eigenvectors:

• For ${\lambda }_1=3$: $(A-3I)\mathbf{v}=0\Rightarrow {\mathbf{v}}_1=(1,2{)}^T$.
• For ${\lambda }_2=-1$: $(A+I)\mathbf{v}=0\Rightarrow {\mathbf{v}}_2=(1,-2{)}^T$.

General solution:

$\mathbf{x}(t)=c_1\left[\begin{array}{c}1\\ 2\end{array}\right]e^{3t}+c_2\left[\begin{array}{c}1\\ -2\end{array}\right]e^{-t}$

The fundamental matrix is

$\Phi (t)=\left[\begin{array}{cc}{e}^{3t}& e^{-t}\\ 2e^{3t}& -2e^{-t}\end{array}\right],\det \Phi (t)=-4e^{2t}\ne 0$

so the two columns are independent for every $t$ — confirming we have a real general solution.

Nonhomogeneous systems

For ${\mathbf{x}}^{\prime }=A\mathbf{x}+\mathbf{b}(t)$, the same decomposition as for scalar ODEs applies: $\mathbf{x}(t)={\mathbf{x}}_c(t)+{\mathbf{x}}_p(t)$ where ${\mathbf{x}}_c$ is the general solution of the homogeneous system and ${\mathbf{x}}_p$ is any particular solution. See Particular solution and complementary solution for the scalar version of the same idea — vector-valued is identical in structure.

In context

The general solution is the bridge between abstract existence/uniqueness theorems and concrete numerical answers. Once you have it, you’ve solved the system forever — every initial condition reduces to picking constants. For the geometric picture of how trajectories arrange themselves, see Phase plane behaviour.