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

Question

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

Accepted Answer

∻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.⚹◊What is the formula for Euler's method in numerical approximation of solutions to differential equations?⍭ Generate me a similar question◊

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