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 $y^\prime$

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 $(1,2)$.