Let $y=f(x)$ be the solution to the differential equation $\frac{d y}{d x}=10 x y-6$ with initial condition $f(6)=1$ . What is the approximation for $f(13)$ obtained by using Euler's method with two step sizes of equal length, starting at $x=6$ ? If necessary, round to three decimal places.

View step-by-step help

Home

Math Problems

Calculus

Euler's method

Full solution

Q. Let

y=f(x)

be the solution to the differential equation

\frac{d y}{d x}=10 x y-6

with initial condition

f(6)=1

. What is the approximation for

f(13)

obtained by using Euler's method with two step sizes of equal length, starting at

x=6

? If necessary, round to three decimal places.

Calculate Step Size: Number of steps = $2$ $\newline$ $x_0 = 6$ and $x = 13$ $\newline$ Find step size $h$ : $\newline$ $h = (13 - 6) / 2$ $\newline$ $h = 7 / 2$ $\newline$ $h = 3.5$ $\newline$ The step size for Euler's method is $3.5$ .
Find $x_1$ : We found: $\newline$ $x_0 = 6$ $\newline$ Step size $h = 3.5$ $\newline$ Find $x_1$ : $\newline$ $x_1 = x_0 + h$ $\newline$ $x_1 = 6 + 3.5$ $\newline$ $x_1 = 9.5$
Calculate $f(x_0, y_0)$ : We found: $\newline$ $f(x, y) = 10xy - 6$ $\newline$ $y_0 = 1$ , $x_0 = 6$ $\newline$ Find $f(x_0, y_0)$ : $\newline$ $f(x_0, y_0) = 10 \times x_0 \times y_0 - 6$ $\newline$ $f(x_0, y_0) = 10 \times 6 \times 1 - 6$ $\newline$ $f(x_0, y_0) = 60 - 6$ $\newline$ $f(x_0, y_0) = 54$
Find $y_1$ : We found: $\newline$ $f(x_0, y_0) = 54$ $\newline$ $y_0 = 1$ , $x_0 = 6$ and $h = 3.5$ $\newline$ Find $y_1$ : $\newline$ $y_1 = y_0 + h \times f(x_0, y_0)$ $\newline$ $y_1 = 1 + 3.5 \times 54$ $\newline$ $y_1 = 1 + 189$ $\newline$ $y_1 = 190$
Calculate $f(x_1, y_1)$ : We found: $\newline$ $f(x, y) = 10xy - 6$ $\newline$ $y_1 = 190$ , $x_1 = 9.5$ $\newline$ Find $f(x_1, y_1)$ : $\newline$ $f(x_1, y_1) = 10 \times x_1 \times y_1 - 6$ $\newline$ $f(x_1, y_1) = 10 \times 9.5 \times 190 - 6$ $\newline$ $f(x_1, y_1) = 10 \times 1805 - 6$ $\newline$ $f(x_1, y_1) = 18050 - 6$ $\newline$ $f(x_1, y_1) = 18044$
Find $y_2$ : We found: $\newline$ $f(x_1, y_1) = 18044$ $\newline$ $y_1 = 190$ , $x_1 = 9.5$ and $h = 3.5$ $\newline$ Find $y_2$ : $\newline$ $y_2 = y_1 + h \times f(x_1, y_1)$ $\newline$ $y_2 = 190 + 3.5 \times 18044$ $\newline$ $y_2 = 190 + 63154$ $\newline$ $y_2 = 63344$
Final Approximation: What is the approximation for $f(13)$ obtained by using Euler's method with two step sizes of equal length, starting at $x=6$ ? $\newline$ The final approximation for $f(13)$ is the value of $y$ after the second step of Euler's method. $\newline$ The value of $y$ after the second step of Euler's method is $63344$ .