Variation of constants

Variation of constants is the method that produces a particular solution of a non-homogeneous linear ODE without guessing any ansatz.

Consider a linear system of first order with constant coefficients (see system of first order linear ODEs- real case)

X^{'} (t) = A X (t) + F (t), A \in R^{n \times n}

where $F (t)$ is a given function. By superposition, the general solution is

X (t) = X_{h} (t) + X_{p} (t)

where $X_{h}$ is the general solution of the homogeneous system $X^{'} = A X$ and $X_{p}$ is one particular solution of the full system. (The homogeneous part is a vector space, see solution space of an ODE.)

Let $ϕ_{1}, \dots, ϕ_{n}$ be $n$ linearly independent solutions of $X^{'} = A X$ , collected as columns in the fundamental matrix

Φ (t) = (\begin{matrix} | & | & | \\ ϕ_{1} & ϕ_{2} & \dots & ϕ_{n} \\ | & | & | \end{matrix})

It satisfies $Φ^{'} = A Φ$ and, by construction, $det Φ (t) \neq 0$ .

Theorem
Let $Φ (t)$ be a fundamental matrix of the homogeneous system $X^{'} = A X$ . Then a particular solution of the non-homogeneous system $X^{'} = A X + F$ is given by

X_{p} (t) = Φ (t) \int Φ^{- 1} (t) F (t) d t

$◼$

Proof
Guess a particular solution with the shape of a homogeneous one, but let the constants vary with $t$ :

X_{p} (t) = c_{1} (t) ϕ_{1} (t) + c_{2} (t) ϕ_{2} (t) = Φ (t) c (t)

Observe that

X_{p}^{'} - A X_{p} = Φ^{'} c + Φ c^{'} - A Φ c = \underset{= 0}{\underset{⏟}{(Φ^{'} - A Φ) c}} + Φ c^{'} = Φ c^{'}

The homogeneous part of the ansatz cancels, exactly because $Φ$ was built out of solutions of $X^{'} = A X$ . What survives is $Φ c^{'}$ , and the requirement $X_{p}^{'} - A X_{p} = F$ becomes the ordinary system

Φ c^{'} = F ⟹ c^{'} = Φ^{- 1} F ⟹ c (t) = \int Φ^{- 1} (t) F (t) d t

Putting $X_{p} = Φ c$ back gives the formula. $◼$

Remark

For

y^{(n)} + p_{n - 1} (t) y^{(n - 1)} + \dots + p_{0} (t) y = r (t)

set $X = (y, y^{'}, \dots, y^{(n - 1)})^{T}$ , so that $X^{'} = A X + F$ with $A$ the companion matrix and $F = (0, \dots, 0, r)^{T}$ . Then $det Φ (t)$ is exactly the Wronskian of the homogeneous basis $ϕ_{1}, \dots, ϕ_{n}$ , and the method is the "variation of parameters" of the scalar literature.