Solving Differential Equations

Graphical Soln's

100

A graphical representation of the solutions of a first-order differential equation

What is a slope field?

200

The curve plotted through a slope field given an initial condition (in the form of a point).

What is the solution curve?

300

A condensed version of a slope field that can be used only for autonomous differential equations.

What is a phase line?

400

The arrows on the phase line both point toward an equilibrium point.

What is a sink?

500

The equilibrium solutions of the DE y'=y²-y-6.

What are y = -2 and y = 3 ?

Euler's Method

100

The change in x used in the table method (usually given).

What is the step size?

200

Use Euler's Method with a step size of 0.2 to estimate y(1.2), where y(x) is the solution to the initial value problem y'=y-2x, with y(1)=0

y=-0.4

300

Use Euler's Method to estimate y(1) using a step size of 0.1, if y'=x+(y/5)

y(1) ≈ -3.182

Separable Variables

100

Is y' = x - xy separable?

Yes

200

The general solution to 

y' = 2y + 1.

What is y = Ce^2x − 1/2?

300

Find the solution to the differential equation y' = sin(x) given the initial condition y(0) = 2.

y = -cos(x)+2

400

Find the general solution to dy + 7x(dx) = 0

y=−3.5x² + C

500

In a population of fixed size S, the rate of change of the number N of persons who have heard a rumor is proportional to the number of those who have not yet heard it. Model this situation with a differential equation.

N' = k(S − N)

First Order

100

These DEs are linear:

y'+2xy=4

y'=sin(x)

y'=3e^xy

y'+2xy=4

y'=sin(x)

200

These types of differential equations can be solved using the method of integrating factors.

What is Linear?

300

This DE is an example of one that could be solved using separable variables and the method of integrating factors.

(Wide variety of DEs possible.)

400

The solution to dy/dt=ky, y(0)=y₀.

What is y=y₀e^kt?

500

The solution to y'-y=x, y(0)=1

What is y=2e^x-x-1?

Second Order

100

These real-life situations can be modeled with a second-order, constant coefficient differential equation.

What are mass-spring systems and circuits?

200

The determinant of a matrix of functions and its derivatives.

What is the Wronskian?

300

The annihilator for 

x³-4x²+10.

What is D⁴ ?

400

The general solution to 

y''-y=0.

What is y=c₁e^x+c₂e^-x ?

500

The general solution to y''+6y'+9y=0.

What is y=c₁e^-3x+c₂xe^-3x ?