Solve. Round your results to four decimal places.
-Use the Euler method with $d x = 0.2$ to estimate $y ( 3 )$ if $y ^ { \prime } = - y / x$ and $y ( 2 ) = 3$. What is the exact value of $y ( 3 )$ ?
A) $\mathrm { y } \approx 0.7895$, exact value is $2.0000$
B) $\mathrm { y } \approx 1.9286$, exact value is $1.5000$
C) $\mathrm { y } \approx 1.1053$, exact value is $6.0000$
D) $y \approx 1.7368$, exact value is $3.0000$

Question

Quizplus · Accepted Answer

The Euler method approximation can be calculated using the given differential equation and initial condition. The exact solution of the differential equation $y' = -y/x$ with the initial condition $y(2) = 3$ is $y = C/x$, where $C$ is a constant. By substituting the initial condition, we find $C = 6$, so the exact solution is $y = 6/x$. Evaluating this at $x = 3$ gives the exact value $y(3) = 2$. The approximation using the Euler method with $dx = 0.2$ and starting from $x = 2, y = 3$ until $x = 3$ will not match the options exactly, but the correct exact value helps identify the correct choice.

Solve Round Your Results to Four Decimal Places dx=0.2d x = 0.2dx=0.2

Solve Round Your Results to Four Decimal Places $d x = 0.2$