MAT244H1 Lecture Notes - Lecture 1: Integral Curve, Product Rule
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January 8, 2019
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
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:
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.