Question

Math

Posted 3 months ago

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

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Answer from Sia

Posted 3 months ago

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.

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