Differential Equations 1

I want my notes are full of Linear Algebra, Differential Equations, Metrics, Geometry, Duals, etc. I just want to prepare students for Relativity. So build upon SolitaryRoad.com documents.

https://solitaryroad.com/de.html

Differential equation

Equation containing derivatives

Differential form

For example, we have \[x\frac{dy}{dx}+3y=y^3\] into \[xdy=(y^3-3y)dx\].

Ordinary differential equation

Notation \(d\)

Partial differential equation

Notation \(\partial\)

Order of a differential equation

The order of the highest ordered derivative

Degree of a differential equation

Highest power of the derivatives

https://byjus.com/maths/differential-equation/

Linear differential equation

Degree is 1. \[a_0(x)\frac{d^ny}{dx^n}+a_1(x)\frac{d^{n-1}y}{dx^{n-1}}+...+a_{n-1}(x)\frac{dy}{dx}+a_n(x)y=f(x)\]

Singular solutions

Singular solutions is an envelope, see previous Differential Geometry notes definition for envelope.