Representation theorem

The representation theorem says that the general solution of a homogeneous linear ODE can be written as a linear combination of $n$ linearly independent solutions, where $n$ is the order of the ODE. Once you have $n$ such solutions (a fundamental set), every solution is a linear combination of them.

For a second-order homogeneous linear ODE $a_2(x)y^{\prime \prime }+a_1(x)y^{\prime }+a_0(x)y=0$:

If $y_1(x),y_2(x)$ are two solutions on an interval $(a,b)$ and at some $x_0\in (a,b)$ the Wronskian satisfies

$W[y_1,y_2](x_0)\ne 0$

then $y_1,y_2$ are linearly independent, and the general solution is

$y(x)=c_1{y}_1(x)+c_2{y}_2(x)$

for arbitrary constants $c_1,c_2$. Every solution of the ODE has this form for some choice of constants.

Why it works

Two ingredients combine to give the theorem:

1. The Superposition principle: any linear combination of solutions is a solution. So $c_1{y}_1+c_2{y}_2$ is always a solution.

2. The dimension count: the solution space of an $n$-th order homogeneous linear ODE is exactly $n$-dimensional. The construction: pick a point $x_0$, and for each standard basis vector ${\mathbf{e}}_k\in {\mathbb{R}}^n$ ($k=0,\dots ,n-1$) build the IVP whose initial conditions are $y(x_0)=e_{k,0}$, $y^{\prime }(x_0)=e_{k,1}$, …, $y^{(n-1)}(x_0)=e_{k,n-1}$. Each of these $n$ IVPs has a unique solution by the Existence and uniqueness theorem (linear ODEs in standard form satisfy the Lipschitz condition globally on the interval where the coefficients are continuous). The $n$ resulting solutions are linearly independent at $x_0$ — their Wronskian there is the determinant of the identity matrix, equal to 1. So they span an $n$-dimensional subspace of solutions. Conversely, any solution is determined by its $n$ values at $x_0$, so it equals the corresponding linear combination of these basis solutions. Hence the solution space has dimension exactly $n$.

If $y_1,y_2$ are linearly independent (Wronskian nonzero), they span the 2D solution space — every solution is a combination.

Why two are enough

A second-order ODE needs two initial conditions ($y(x_0)$ and $y^{\prime }(x_0)$). Given $y(x)=c_1{y}_1+c_2{y}_2$, the system

$\{\begin{array}{l}{c}_1{y}_1(x_0)+c_2{y}_2(x_0)=y_0\\ c_1{y}_1^{\prime }(x_0)+c_2{y}_2^{\prime }(x_0)=y_0^{\prime }\end{array}$

is a $2\times 2$ linear system for $c_1,c_2$. The coefficient matrix is exactly the Wronskian matrix. If $W\ne 0$, the system has a unique solution — every IVP can be solved by adjusting $c_1,c_2$. If $W=0$, some IVPs can’t be solved by linear combinations of $y_1,y_2$, meaning these two don’t span the solution space.

Worked example

$y^{\prime \prime }+y=0$ has solutions $y_1=\cos x$, $y_2=\sin x$.

Wronskian: $\det \left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right]={\cos }^{2}x+{\sin }^{2}x=1\ne 0$.

So $y_1,y_2$ are linearly independent everywhere, and the general solution is

$y(x)=c_1\cos x+c_2\sin x$

For initial conditions $y(0)=a$, $y^{\prime }(0)=b$: $c_1=a$, $c_2=b$. So $y(x)=a\cos x+b\sin x$.

The general $n$-th order version

For an $n$-th order homogeneous linear ODE, the solution space is $n$-dimensional. The general solution is

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

with $\{y_1,\dots ,y_n\}$ any set of $n$ linearly independent solutions. The Wronskian extends to $n\times n$:

$W[y_1,\dots ,y_n](x)=\det \left[\begin{array}{cccc}{y}_1& y_2& \cdots & y_n\\ y_1^{\prime }& y_2^{\prime }& \cdots & y_n^{\prime }\\ ⋮& ⋮& & ⋮\\ y_1^{(n-1)}& y_2^{(n-1)}& \cdots & y_n^{(n-1)}\end{array}\right]$

If $W\ne 0$ at some point, the solutions are independent and span the solution space.

In context

The representation theorem is the foundation of how we solve linear ODEs: find $n$ linearly independent solutions (via Characteristic equation for constant coefficients, or other methods for variable coefficients), then write the general solution as their linear combination.

For nonhomogeneous ODEs, the general solution adds a particular solution: $y=y_c+y_p$ where $y_c$ is the homogeneous part (covered by this theorem) and $y_p$ is any specific solution of the nonhomogeneous equation. See Particular solution and complementary solution.

For systems of linear ODEs, an analogous theorem applies — see General solution of linear system.