Higher Order ODEs

Linear Differential Equations

\frac{\text{d}^ny}{\text{d}x^n}+ p_1(x)\frac{\text{d}^{n-1}y}{\text{d}x^{n-1}}+ \;...\; +p_n(x)y =q(x)

Based on $q(x)$ , the above equation is categorized into $2$ types:

Homogenous if $q(x)=0$
Non-homogenous if $q(x)\not=0$

Solution

The general solution of the equation is $y=y_p+y_c$ .

Here

y=y_p+y_c

Here:

$y_p$: particular solution
$y_c$: complementary solution

Particular Solution

Doesn’t exist for homogenous equations. For non-homogenous equations check steps section of 2nd order ODE.

Complementary Solution

Solutions assuming $LHS=0$ (as in a homogenous equation).

y_c = \sum_{i=1}^{n}{c_i\,y_i}

Here:

$c_i$: constant coefficients
$y_i$: a linearly-independent solution

Linearly Dependent & Independent

$n$ -th order linear differential equations have n linearly independent solutions.

Two solutions of a differential equation $u,v$ are said to be linearly dependent, if there exists constants $c_1,c_2\;(\not=0)$ such that $c_1u(x)+c_2v(x)=0$ .

Otherwise, the solutions are said to be linearly independent, which means:

\sum_{i=1}^{n}{c_iy_i}=0\implies \forall{c_i}=0

Linear Differential Operators with Constant Coefficients

Differential Operator

Defined as:

\text{D}^i=\frac{\text{d}^i}{\text{d}x^i}\;;\;n\in\mathbb{Z}^+

The above equation can be written using the differential operator:

\text{D}^ny+ a_1\text{D}^{n-1}y+ \;...\; +a_ny =q(x)

Here if $y$ is factored out (how tf?):

(\text{D}^n+ a_1\text{D}^{n-1}+ \;...\; +a_n )y = P(D)y = q(x)

where $P(D)=(\text{D}^n+a_1\text{D}^{n-1}+\;...\;+a_n)$ .

$P(D)$ is called a polynomial differential operator with constant coefficients.

Written by

Sahithyan K.

November 3, 2026 1 min read

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