9/12 ODE Rolling Notes

Logistics

Office hours:
- Wednesday 11:00am - 12:00pm, MLC
- Wednesday 12:00pm - 1:00pm, Van Vleck 407

Beginning of e-textbook has 2 tables of differentiation and integration rules

There will be about 6 quizzes given in discussion section, about 10-15 minutes long
Next quiz will be next week, covering Section 1.1, 1.2

Content

Sections 1.1 (contd) and 1.2 -> Contents and initial value problems

Warm-up

Which of the following are solutions to dxdt=−x2\frac{dx}{dt} = -x^2?
I: x(t)=0x(t) = 0 on (−∞,∞)(-\infty, \infty)
II: x(t)=1tx(t) = \frac{1}{t} on (−∞,∞)(-\infty, \infty)
III: x(t)=1tx(t) = \frac{1}{t} on (0,∞)(0, \infty)

Answer: I and III*
We see that II isn't a solution because x(0)x(0) is undefined.

Differential equations relate how quantities change over time.
Differential equation goals:
- Use differential equations as mathematical models for natural phenomena
- Find solutions to differential equations (exact or approximate)
- Interpret the solutions - what can we learn?

Linearity

Definition: Linearity

An n-th order ODE is linear if it can be written in the following form:
an(x)dnydxn+an−1(x)dn−1ydxn−1+⋯+a1(x)dydx+a0(x)y=g(x)a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x)\frac{dy}{dx} + a_0(x)y = g(x)

Notes:
- y and its derivatives appear only as linear terms (every y or derivative only appear as first-degree terms)
- an(x),…,a0(x)a_n(x), \dots , a_0(x) and g(x)g(x) can be arbitrary functions of xx

An ODE is nonlinear if it is not linear (of course).

We like linear ODEs because we have a (relatively) easy strategy to solve them.
We can only take integrals of specific things, and we solve ODEs by solving integrals. We have a good way of taking integrals of linear ODEs.

Activity

Determine the order of the following ODEs. Are they linear?
1) dPdt=P(1−P)\frac{dP}{dt} = P(1 - P)
2) x3y′′+2x2y′−xy=12x2x^3 y'' + 2x^2 y' - xy = 12x^2
3) 2d3ydx3=ycos⁡(x)2\frac{d^3 y}{dx^3} = y\cos{(x)}

The order of (1) is 1, as we take P to its first derivative. It isn't linear, because from simplifying we get a P2P^2 form.
The order of (2) is 2, as we see a y′′y'' term. It is linear, because each y-term is a n-derivative of y.
The order of (3) is 3, because we have a 3rd derivative of y. It is linear, because all terms of y are n-derivatives of y (the cos⁡(x)\cos{(x)} doesn't matter because it's a function of x).

More about Solutions

Definition: Solutions to N-th Order Odes

A solution to an n-th order ODE F(x,y,y′,…,y(n))=0F(x, y, y', \dots , y^(n)) = 0 is a function ϕ(x)\phi(x) where
- ϕ,ϕ′,…,ϕ(n)\phi, \phi', \dots, \phi^(n) are continuous on some interval II
- F(x,ϕ,ϕ′,…,ϕ(n))=0F(x, \phi, \phi', \dots, \phi^(n)) = 0 (ϕ\phi satisfies the equation)

If ϕ=0\phi = 0 is a solution, it is called a trivial solution.
A graph of a solution ϕ\phi is called a solution curve.

An explicit solution is a solution where we have solved for ϕ\phi explicitly.
Example of implicit solution: x2+y2=25x^2 + y^2 = 25 is an implicit solution to dydx=−xy\frac{dy}{dx} = -\frac{x}{y}.
We can use implicit differentiation to validate this implicit solution. We can solve for y to find explicit solutions as well.

Our strategies for solving ODEs typically involve taking intervals. This results in constants of integration.
Typically, an n-th order ODE has infinitely many solutions. In fact, an n-parameter family of solutions. #-> because there are n constants from integrating n times?
A specific solution with no parameters is called a particular solution. E.g. initial value problems