Rongzhou at 20:33, 10 October 2026

2026-10-10T20:33:06Z

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

	{{#content:Q2136}}		{{#content:Q2136}}

Rongzhou: Created page with "A linear differential equation of order $n$ has the general form {{#content:Q2136}} 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..."

2026-10-10T20:32:05Z

Created page with "A linear differential equation of order $n$ has the general form {{#content:Q2136}} 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..."

New page

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

{{#content:Q2136}}

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

Linear differential equation - Revision history

Rongzhou at 20:33, 10 October 2026