Module 5 First Order Differential Equations

Flashcards for Module 5 First Order Differential Equations Lecture

What is a differential equation (DE)?

Contains derivatives of an unknown function.

What is the difference between algebraic equations and differential equations?

Algebraic equations do not contain derivatives of an unknown function.

What does the order of a DE refer to?

The highest derivative in the equation.

Give an example of a 1st order DE.

y' = xy + 2x

Give an example of a 2nd order DE.

y'' + 2y' + 3 = x

Give an example of a 3rd order DE.

y''' = 2y' - x

What can direction fields be used for?

To understand and visualize the behavior of solutions to a DE.

What is the general form of a 1st order DE that we will consider?

dy/dx = f(x,y)

How is a direction field generated?

Plotting slope lines at an array of points.

To plot a solution on a direction field, what must you do after specifying a starting point (x0, y0)?

Move from that starting point in a direction parallel to the surrounding field lines.

When plotting solutions on a direction field, what is an important rule to remember?

You cannot cross a slope line if you are moving parallel to it.

What is an initial value problem (IVP)?

A DE with a starting point.

What form does a first order autonomous DE have?

dy/dx = f(y)

How does the direction field change for an autonomous vs non autonomous DE?

Autonomous DES slopes do not depend horizontally on X.

What is an equilibrium solution of an autonomous DE dy/dx = f(y)?

A solution of the form y = constant (flat).

How do we find equilibrium solutions?

Solving f(y) = 0.

When is an equilibrium solution y = c stable?

All nearby solutions approach c as the independent variable (x) -> infinity.

When is an equilibrium solution y = c unstable?

All nearby solutions move away from c as the independent variable (x) -> infinity.

How can stability be determined?

Using a sign diagram.

What is the equation for natural growth?

dP/dt = rP

What is the equation for logistic growth?

dP/dt = rP(1 - P/k)

What does k stand for in dP/dt = rP(1 - P/k)?

k = carrying capacity.

What equation expresses Newton's 2nd law, with air resistance?

mg - kv = ma

What is the equation for terminal velocity?

V = gm/k

What is form called when a 1st order DE is separable?

dy/dx = f(y)g(x)

What is the solution technique for separable equations?

Integrate both sides after separating variables: ∫dy/f(y) = ∫g(x) dx.

What is the explicit solution to dy/dx = xy?

y = A e^(x^2/2)

What is it called when all the solutions to the DE have the form y = A e^(x^2/2) ?

General solution.

How can we get a particular solution?

By specifying an initial condition (IC).

What is the standard form of a 1st order linear DE?

dy/dx + P(x)y = Q(x)

What is step 1 in solving a linear DE?

Calculate the integrating factor: I(x) = e^(∫P(x) dx)

What is step 2 in solving a linear DE?

Evaluate ∫I(x)Q(x) dx

What is step 3 in solving a linear DE?

Combine into the general solution form: y = (1/I(x)) [∫I(x)Q(x) dx + C]

How do you determine a value for C for a linear equation?

By specifying an IC to give an IVP.

For linear DEs, how do you determine the domain of the solution to an IVP?

Consider where P(x) and Q(x) are continuous.

What theorem describes the existence and uniqueness of a solution?

If P & Q are continuous on an interval (a, b) containing x0, exists a unique function y(x) that satisfies the DE for all x in (a, b) & that also satisfies the IC y(x0) = y0.