# MAT244H1 Lecture Notes - Lecture 1: Integral Curve, Product Rule

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Lecture 1

January 8, 2019

1 Introduction

A Diﬀerential Equation (DE) involves a function and its derivative, its order is

the highest power of the function. The expression:

F(x, y, y′, y′′ , ..., yn) is an Ordinary Diﬀerential Equation(ODE) of degree n.

Where xis an independent variable and yis the solution. This is called the

integral curve y(x) trajectory. ODE treats a single variable, while PDE stands

for parital meaning ∂y

∂x1,∂y

∂x2, ...

Sometime the ODE is in normal form y(x) = G(x, y, y′, ..., yn−1). We say tha

the ODE is linear if F is linear in the variables y, ..., ynotherwise it is non-linear.

All nth order ODEs which are linear have the form:

an(t)yn+an−1(t)yn−1+... +a1(t)y′+a0(t)y=g(t)

Examples:

1. y′′ + 4y′+ 5y= sinx this is a 3rd linear order ODE

2. F(x, y, y′, y′′ , y′′′ ) = y′′′ + 2exy′′ +yy′x4this is a non linear ODE

Deﬁnition: an(t)yn+an−1(t)yn−1+... +a1(t)y′+a0(t)y=g(t) is homo-

geneous if g(t) = 0, otherwise it is non-homogeneous.

Deﬁnition: A function E is linear if:

1. E(αx) = αE(x)

2. E(α+x) = E(x) + E(α)

Deﬁnition: We say a continuous function is a solution to the ODE F(x, y, y′, ..., yn)

on an interval I if the derivatives u′, u′′ , ..., unexist ∀xin I.

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