use eulers method with (h = 0.2) to approximate (x(0.4)) and (y(0.4)), where (x(t), y(t)) is the solution of the initial-value problem

\
\begin{aligned}
x &= x + y \\
y &= x - y \\
x(0) &= 1, y(0) = 2.
\end{aligned}
\

(x(0.2) =)
(y(0.2) =)
(x(0.4) =)
(y(0.4) =)

Question

use eulers method with (h = 0.2) to approximate (x(0.4)) and (y(0.4)), where (x(t), y(t)) is the solution of the initial-value problem

\
\begin{aligned}
x &= x + y \\
y &= x - y \\
x(0) &= 1, y(0) = 2.
\end{aligned}
\

(x(0.2) =)
(y(0.2) =)
(x(0.4) =)
(y(0.4) =)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Euler's Method for Systems", "Numerical Methods for ODEs" ], "current_concepts": [ "Euler's Method for Systems", "Numerical Methods for ODEs", "Initial Value Problems for Systems" ] } ## State the system and initial conditions $$ \begin{aligned} &x' = f(t, x, y) = x + y \ &y' = g(t, x, y) = x - y \ &t_0 = 0, \quad x_0 = x(0) = 1, \quad y_0 = y(0) = 2, \quad h = 0.2 \end{aligned} $$ ## Calculate the first step at t = 0.2 $$ \begin{aligned} x_1 &= x_0 + h \cdot f(t_0, x_0, y_0) = 1 + 0.2 \cdot (1 + 2) = 1 + 0.6 = 1.6 \ y_1 &= y_0 + h \cdot g(t_0, x_0, y_0) = 2 + 0.2 \cdot (1 - 2) = 2 - 0.2 = 1.8 \end{aligned} $$ ## Calculate the second step at t = 0.4 $$ \begin{aligned} x_2 &= x_1 + h \cdot f(t_1, x_1, y_1) = 1.6 + 0.2 \cdot (1.6 + 1.8) = 1.6 + 0.2 \cdot 3.4 = 1.6 + 0.68 = 2.28 \ y_2 &= y_1 + h \cdot g(t_1, x_1, y_1) = 1.8 + 0.2 \cdot (1.6 - 1.8) = 1.8 + 0.2 \cdot (-0.2) = 1.8 - 0.04 = 1.76 \end{aligned} $$ Use Euler's method with $h = 0.2$ to approximate $x(0.4)$ and $y(0.4)$, where $x(t), y(t)$ is the solution of the initial-value problem $$ \begin{aligned} x' &= x + y \ y' &= x - y \ x(0) &= 1, \quad y(0) = 2. \end{aligned} $$ $x(0.2) =$ 1.6 $y(0.2) =$ 1.8 $x(0.4) =$ 2.28 $y(0.4) =$ 1.76 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Euler's Method for Systems" ] }

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QUESTION IMAGE

Question

Explanation:

State the system and initial conditions

Calculate the first step at t = 0.2

Calculate the second step at t = 0.4

Answer: