Question

Math

Posted 6 months ago

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2. Let $y=f(x)$ be the solution to the differential equation $\frac{d y}{d x}=1+2 y$ with the initial condition $f(0)=1$. What is the approximation for $f(1)$ if Euler's method is used, starting at $x=0$ with a step size of 0.5 ?
(A) 2.5
(B) 3.5
(C) 4.0
(D) 5.5
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 1

Apply Euler's method to the differential equation $\frac{dy}{dx} = 1 + 2y$ with initial condition $y(0) = 1$ and a step size of 0.5

step 2

Calculate the value of $y$ at $x = 0.5$. Use the formula $y_{next} = y_{current} + step\_size \cdot \frac{dy}{dx}|_{x_{current}, y_{current}}$

step 3

For the first step, $y_{current} = 1$ and $\frac{dy}{dx}|_{x_{current}, y_{current}} = 1 + 2 \cdot 1 = 3$. Thus, $y_{next} = 1 + 0.5 \cdot 3 = 2.5$

step 4

Now calculate the value of $y$ at $x = 1$. Again use the formula from step 2 with the new $y_{current} = 2.5$ and $\frac{dy}{dx}|_{x_{current}, y_{current}} = 1 + 2 \cdot 2.5 = 6$

step 5

For the second step, $y_{next} = 2.5 + 0.5 \cdot 6 = 5.5$

Answer

(D) 5.5

Key Concept

Euler's Method for Approximating Solutions to Differential Equations

Explanation

Euler's method uses the slope of the tangent line at known points to estimate the value of the function at the next point, using a given step size.

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