Semester 3 / Differential Equations /Introduction

Introduction to Differential Equations

1 min read Updated Fri Apr 24 2026 03:46:02 GMT+0000 (Coordinated Universal Time)
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Different methods of solving differential equations, especially partial differential equations are dicussed here.

Definitions

Even function

When a function is symmetric about the y-axis.

∀x∈Dom(f) f(−x)=f(x)\forall x \in \text{Dom}(f)\, f(-x) = f(x)

Odd function

When a function is skew-symmetric about the y-axis.

∀x∈Dom(f) f(−x)=−f(x)\forall x \in \text{Dom}(f)\, f(-x) = -f(x)

Periodic function

ff is periodic iff:

∃p>0 ∀x∈Dom(f) f(x+p)=f(x)\exists p \gt 0\, \forall x \in \text{Dom}(f)\, f(x+p) = f(x)

The smallest pp satisfying the above realtionship is called the period. The function is also called as p-periodic.

Orthogonal

2 distinct functions, ff and gg, are orthogonal over the interval [a,b][a, b] (or (a,b)(a,b) or (a,b](a,b] or [a,b)[a,b)) iff:

∫abf(x)g(x) dx=0\int_a^b f(x)g(x)\,\text{d}x = 0

A set of functions fif_i are orthogonal over the interval [a,b][a, b] (or (a,b)(a,b) or (a,b](a,b] or [a,b)[a,b)) iff:

∫abfi(x)fj(x) dx=0fori≠j\int_a^b f_i(x)f_j(x)\,\text{d}x = 0 \quad \text{for} \quad i \neq j