Higher Order Ordinary Differential Equations

Linear Differential Equations

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

• $y_p$ - 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). $$

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:

Linear differential operators with constant coefficients

Differential operator

Defined as:

The above equation can be written using the differential operator:

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

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.$$