Linear differential equation

A linear differential equation of order $n$ has the general form

$$\sum_{|\alpha|\le m} a_\alpha(\mathbf{x})\,D^\alpha u(\mathbf{x})=f(\mathbf{x}), \qquad \mathbf{x}=(x_1,\dots,x_d).$$

Where:

$a_\alpha(\mathbf{x})$ and $f(\mathbf{x})$ are functions of $\mathbf{x}$;

$$ \alpha=(\alpha_1,\dots,\alpha_d), \qquad |\alpha|=\alpha_1+\cdots+\alpha_d; $$

and

$$ D^\alpha u= \frac{\partial^{|\alpha|}u} {\partial x_1^{\alpha_1}\cdots \partial x_d^{\alpha_d}}. $$

When $f(\mathbf{x})=0$, the differential equation is further said to be homogeneous.

Properties

Superposition

Consider the linear differential equation

$$ \sum_{|\alpha|\le m} a_\alpha(\mathbf{x})\,D^\alpha u(\mathbf{x})=f(\mathbf{x}), $$

as a linear operator $L[u]$,

by definition of linearity

$$ L[c_1u_1+c_2u_2\cdots +c_nu_n]=c_1L[u_1]+c_2L[u_2]+\cdots c_nL[u_n], $$

Homogeneous case

If $L[u]=f(\mathbf{x})=0$

Given

$$ L[u_1], L[u_2]\cdots ,L[u_n]=0 $$

Then

$$ L[c_1u_1+c_2u_2\cdots +c_nu_n]=c_1L[u_1]+c_2L[u_2]+\cdots c_nL[u_n]=0, c \in \mathbb{F} $$

Therefore, if solution functions $u_1,u_2\cdots ,u_n$ have been identified for homogeneous linear differential equation $L[u]$, it has general solution $u_h=c_1u_1+c_2u_2\cdots +c_nu_n$.

Non-homogeneous case

When $L[u]=f(\mathbf{x}) \neq 0$,

$$ L[u_h]=0,\quad L[u_p]=f(\mathbf{x}) \Longrightarrow L[u_h+u_p]=0+f(\mathbf{x})=f(\mathbf{x}) $$

Hence the general solution of a non-homogeneous linear differential equation $L[u]=f(\mathbf{x})\neq 0$ is

$$ u=u_h+u_p: $$

where $u_h$ is the general solution of its homogeneous counterpart $L[u]=0$, and $u_p$ is a particular solution to $L[u]=f(\mathbf{x})$