Higher Order Ordinary Differential Equations

Linear Differential Equations

Based on , the above equation is categorized into types:

• Homogenous if
• Non-homogenous if

For 1st semester

Only linear, ordinary differential equations with constant coefficients are required.

They can be written as:

Solution

The general solution of the equation is .

Here

• - particular solution
• - 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 (as in a homogenous equation).

Here

• - constant coefficients
• - a linearly-independent solution

Linearly dependent & independent

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

Two solutions of a differential equation are said to be linearly dependent, if there exists constants such that .

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

Linear differential operators with constant coefficients

WTF?

I don’t understand anything in this section.

Differential operator

Defined as:

We can write the above equation using the differential operator:

Here if we factor out (how tf?), we get:

where .

is called a polynomial differential operator with constant coefficients.