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

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Published on 1 May 2019
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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, ..., yn1). 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+an1(t)yn1+... +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 +yyx4this is a non linear ODE
Denition: an(t)yn+an1(t)yn1+... +a1(t)y+a0(t)y=g(t) is homo-
geneous if g(t) = 0, otherwise it is non-homogeneous.
Denition: A function E is linear if:
1. E(αx) = αE(x)
2. E(α+x) = E(x) + E(α)
Denition: 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.
1
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