Eulers Method Slope Fields Separable Differential Equations

Euler's Method

100

Use Euler's Method to approximate y(1.1) given dy/dx=x+y with y(1)=2y and step size h=0.1.

y(1.1) ≈ 2.3

200

Use Euler's Method to approximate 𝑦(0.2) given 𝑑𝑦/𝑑𝑥 = 𝑦 − 𝑥  with 𝑦(0) = 1 and step size ℎ= 0.1.

y(0.2)≈1.2

300

Use Euler's Method to approximate y(2.1 given dy/dx=x²−y with y(2)=1 and step size h=0.1

y(2.1)≈1.611

400

Use Euler's Method to approximate y(1.5) given dy/dx=sin⁡(x)−y with y(1)=0 and step size h=0.25

y(1.5)≈0.3947

500

Use Euler's Method to approximate y(1.1) given dy/dx=xy with y(1)=2 and step size h=0.05.

y(1.1)≈2.3318

Slope Fields

100

What is a slope field?

A slope field (or direction field) is a graphical representation of the solutions of a first-order differential equation. It consists of short line segments or arrows that show the slope of the solution curve at various points.

200

Given the differential equation dy/dx=x−y, what is the slope of the line segment at the point (1, 2) in the slope field?

dy/dx= −1

300

Sketch a small portion of the slope field for the differential equation dy/dx=x+y at the points (0, 0), (1,0), and (0,1).

• At (0,0), dy/dx=0 (horizontal line segment)
• At (1,0), dy/dx=1 (line segment with slope 1)
• At (0,1) (dy/dx=1 (line segment with slope 1)

400

Consider the differential equation dy/dx=sin⁡(x). Describe the general pattern you would expect to see in the slope field.

The slopes will oscillate between -1 and 1 as sin⁡(x) oscillates. At x=0,pi ,2pi, the slopes will be 0 (horizontal line segments).

500

Given the differential equation dy/dx= y^2 - x^2, sketch the slope field near the origin. Describe the general behavior of the solution curves in the regions where y^2 > x^2 and y^2 < x^2

• Inside the "parabolic" region defined by y^2 > x^2, the slopes are positive, and the solution curves tend to rise.
• Outside the "parabolic" region defined by y^2 < x^2, the slopes are negative, and the solution curves tend to drop.
• Along the lines y=x and y=−x, the slopes are horizontal, representing equilibrium lines.

SDE's

100

Solve the differential equation dy/dx=2x with the initial condition y(0)=3.

y= x² +3

200

Solve the differential equation dy/dx=y with the initial condition y(1)=2.

y= 2e^ x−1

300

Solve the differential equation dy/dx=x/y with the initial condition y(1)=2

y= sqrt (x²+3)

400

Solve the differential equation dy/dx=sin⁡(x)cos⁡(y) with the initial condition y(0)=pi/2

−cos(0)+C

500

Solve the differential equation dy/dx= x²/y² with the initial condition y(1)=1

y=x

Miscellaneous

100

Evaluate the integral ∫x² dx

x³/3 +C

200

lim x→0    x/sin(5x).

5

300

Solve the differential equation dy/ dx=3y with the initial condition y(0)=2

Answer:

y=2e^3x

400

d/dx  x^x

x^x (1 + ln(x))

500

Find the length of the curve 

 y=ln(cos x) from x=0 to x= pi/4

ln(sqrt 2+1)

Very General Trivia

100

According to an urban legend, what type of animal caused the Great Chicago Fire of 1871 by knocking over a lantern?

Answer: Cow

200

Throughout the Harry Potter novels, Harry is mentioned by name 18,956 times. Which character is the second most mentioned coming in at a total of 6,464?

Answer: Ron Weasley

300

The world capital cities of Vienna, Bratislava, and Budapest all lie along what river, the second-longest in Europe?

Answer: Danube

400

What "G" instrument is technically a metallophone because its set of tuned keys are made of metal plates or tubes rather than the wood of a xylophone? The instrument was featured famously in many Rush songs such as "The Spirit of Radio" and "Closer to the Heart."

Answer: Glockenspiel

500

Only one of the seven countries that borders India has a population with less than one million people. This country famously measures "gross national happiness" and has Thimphu as its capital city. What is this country?

Answer: Bhutan