11-7-25 Euler Method, Slope Field, Implicit Solution

Euler’s Method

How to use Euler’s Method:

  • Given y^\prime = F(x,y),        (x_o,y_o),        step size h

    • x_n = x_{n-1} + h

    • y_n = h\cdot F(x_{n-1},y_{n-1}) + y_{n-1}

Example:

Use Euler’s method with step size 0.1 to estimate y(0.3) where y(x) is the solution to the initial value problem

  • y^\prime = x+y

  • y(0) = 1

Gather)

  • h=0.1

  • x=0.3

  • x_o = 0

  • y_o = 1

  • y= ??

Analyze)

Use the formula:

  • x_1 = x_o + h = 0 + 0.1 = 0.1

  • y_1 = h \cdot F(x_{n-1},y_{n-1}) + y_{n-1} = 0.1 \cdot F(0,1) + 1

    • F(x,y) is the equation given above (y^\prime = x+y), so F(x+y) = x+y = 0+1=1

    • y_1 = 0.1 \cdot 1 + 1 = 1.1

  • x_2 = x_1 + h = 0.1 + 0.1 = 0.2

  • y_2 = h \cdot F(0.1,1.1) + 1.1 = 0.1 \cdot (0.1 + 1.1) + 1.1 = 1.22

  • x_3 =  x_2 + h = 0.2 + 0.1 = 0.3

  • y_2 = h \cdot F(x_2,y_2) + y_2 = 0.1 \cdot (0.2 + 1.22) + 1.22 = 1.362

Now that have x at 0.3 in the step above, we can approximate y(0.3)

  • y(0.3) is approximately 1.362

Slope Field

  • Slope symbol: m

    • When m =:

    • 0 =  \rightarrow

    • \infty or \frac 10 = \uparrow

    • -\infty or -\frac 10 =  \downarrow

Example:

y^\prime = x

Step 1) Make a table of x,y, and y^\prime

*In this example, y is crossed out since it is not used in the equation

Who cares, you learned about this in calc I

Example: Given the equation x² + y² = 100, find \frac {dy}{dx}. Calculate the slope at the point (6,8)

\frac d{dx} (x²+y²=100) \Rightarrow 2x + 2y \frac {dy}{dx} = 0

\Rightarrow 2y \frac {dy}{dx} = -2x \Rightarrow \frac {dy}{dx} = -\frac {2x}{2y} = -\frac xy

 

\frac {dy}{dx} = -\frac 68 = -\frac {(3\cdot 2)}{(4\cdot 2)} = -\frac 34

The slope at (6,8) is -\frac 34 

Example: Given the equation x^3 + 4xy + y² = 13, find \frac {dy}{dx} at the point (1,2).