$$
\frac{d y}{d x}=x-2 y+2 \text { and } f(0)=2 \text {. }
$$

Use Euler's method with two steps of equal size to approximate $f(2.4)$.
$$
f(2.4) \approx
$$

Question

$$
\frac{d y}{d x}=x-2 y+2 \text { and } f(0)=2 \text {. }
$$

Use Euler's method with two steps of equal size to approximate $f(2.4)$.
$$
f(2.4) \approx
$$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To use Euler's method, we first need to determine the step size, $h$. Since we are approximating $f(2.4)$ and starting at $x=0$, with 2 steps, $h = \frac{2.4 - 0}{2} = 1.2$. ‖ ‖ step 2 ⋮ Now we apply the first step of Euler's method using the initial condition $f(0)=2$. We calculate the slope at the initial point using the differential equation $\frac{dy}{dx}=x-2y+2$. At $x=0$, the slope is $0 - 2(2) + 2 = -2$. ‖ ‖ step 3 ⋮ We update the value of $y$ using the slope and the step size: $y_1 = y_0 + h \cdot slope = 2 + 1.2 \cdot (-2) = 2 - 2.4 = -0.4$. The new point to use for the next step is $(x_1, y_1) = (1.2, -0.4)$. ‖ ‖ step 4 ⋮ For the second step, we calculate the slope at $(x_1, y_1) = (1.2, -0.4)$ using the differential equation. The slope is $1.2 - 2(-0.4) + 2 = 1.2 + 0.8 + 2 = 4$. ‖ ‖ step 5 ⋮ We update the value of $y$ again using the new slope: $y_2 = y_1 + h \cdot slope = -0.4 + 1.2 \cdot 4 = -0.4 + 4.8 = 4.4$. The approximate value of $f(2.4)$ is $y_2 = 4.4$. ‖ ∻Answer∻ ⚹ $f(2.4) \approx 4.4$ ⚹ ∻Key Concept∻ ⚹Euler's Method for Approximating Solutions to Differential Equations⚹ ∻Explanation∻ ⚹Euler's method approximates the solution to a differential equation by taking a finite number of steps of a given size, using the slope of the tangent line at the beginning of each step to estimate the increase or decrease in the function's value.⚹◊What is the formula for Euler's method when approximating the value of a function at a specific point?⍭ Generate me a similar question◊

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