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 14.8. 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 15.1.
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.
Theorem15.7.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
where \(C_1, C_2, \dots , C_n\) are arbitrary constants. Technically, the space of all solutions forms an \(n\)-dimensional vector space, see Section 14.1.
Theorem15.8.Linear Inhomogeneous ODEs: Form of General Solution.
The general solution of the \(n^{th}\) order linear inhomogeneous differential equation \(\LL(y)=b(x)\) is
where \(y_{h}\) is the general solution of the homogeneous equation \(\LL(y)=0\) and \(y_{p}\) is any one particular solution of the inhomogeneous equation. Recall that the general solution of the homogeneous equation (above) has \(n\) independent parameters.
Theorem15.9.Linear ODEs: What Initial Conditions Make the Solution Unique.
Consider the \(n^{th}\) order, linear, ordinary differential equation:
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 \(I\text{.}\)
