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Section 8.6 Theorems about Linear ODEs
In this section, we will give, without proof, several important theorems about linear differential equations. But before we get to the theorems, you will need to understand what is meant by the word linear, see Section 8.5 and Section 9.6. so that you can understand the content of these theorems. Before you read this section, you should also make sure you know the definitions and notation in Section 8.1.
The first two theorems tell us the general form for solutions for homogeneous and inhomogeneous linear differential equations and describe the free parameters that exist in each solution. The third theorem describes what kind of initial conditions are necessary to remove this freedom and specify a unique solution.
Theorem 8.6.1. Linear Homogeneous ODEs: Form of General Solution.
An \(n^{th}\) order linear homogeneous differential equation \(\LL(y)=0\) always has \(n\) linearly independent solutions, \({y_1,\dots ,y_n}\text{.}\) The general solution \(y_h\) is
\begin{equation}
y_{h}=C_1 y_1+C_2 y_2 + \dots +C_n y_n\tag{8.6.1}
\end{equation}
where \(C_1, C_2, \dots , C_n\) are arbitrary constants.
Theorem 8.6.2. Linear Inhomogeneous ODEs: Form of General Solution.
The general solution of the \(n^{th}\) order linear inhomogeneous differential equation \(\LL(y)=b(x)\) is
\begin{equation}
y=y_p+y_h\tag{8.6.2}
\end{equation}
where \(y_{h}\) is the general solution of the homogeneous equation \(\LL(y)=0\) and \(y_{p}\) is any particular solution of the inhomogeneous equation. Recall that the general solution of the homogeneous equation (above) has \(n\) indepdendent parameters.
Theorem 8.6.3. Linear ODEs: What Initial Conditions Make the Solution Unique.
Consider the \(n^{th}\) order, linear, ordinary differential equation:
\begin{equation}
1y^{(n)}+a_{n-1}(x)y^{(n-1)}+\dots+a_0(x)y=b(x)\tag{8.6.3}
\end{equation}
together with the initial conditions:
\begin{align}
y(x_0)\amp = c_0\tag{8.6.4}\\
y'(x_0)\amp = c_1\tag{8.6.5}\\
\amp \vdots\notag\\
y^{(n)}(x_0)\amp =c_n\tag{8.6.6}
\end{align}
If \(b(x)\) and \(a_j(x)\) are continuous \(\forall j\) on some interval \(I\) containing \(x_0\text{,}\) then the initial value problem has a unique solution throughout \(I\text{.}\)
