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

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start with the differential equation $\frac{d y}{d x} = 1 + 2y$ and the initial condition $f(0) = 1$. ‖ ‖ step 2 ⋮ Using Euler's method, we approximate the solution at $x = 1$ with a step size of $h = 0.5$. ‖ ‖ step 3 ⋮ The formula for Euler's method is $y_{n+1} = y_n + h \cdot f(x_n, y_n)$. ‖ ‖ step 4 ⋮ At $x = 0$, $y_0 = 1$. We calculate $y_1$ using $y_1 = y_0 + h \cdot (1 + 2y_0)$. ‖ ‖ step 5 ⋮ Substituting the values, $y_1 = 1 + 0.5 \cdot (1 + 2 \cdot 1) = 1 + 0.5 \cdot 3 = 1 + 1.5 = 2.5$. ‖ ‖ step 6 ⋮ Now, we calculate $y_2$ at $x = 1$ using $y_2 = y_1 + h \cdot (1 + 2y_1)$. ‖ ‖ step 7 ⋮ Substituting the values, $y_2 = 2.5 + 0.5 \cdot (1 + 2 \cdot 2.5) = 2.5 + 0.5 \cdot 6 = 2.5 + 3 = 5.5$. ‖ ∻Answer∻ ⚹ 5.5 ⚹ ∻Key Concept∻ ⚹Euler's Method⚹ ∻Explanation∻ ⚹Euler's method is a numerical technique for solving ordinary differential equations by approximating solutions at discrete points.⚹◊What is the formula for Euler's method for approximating a solution to a differential equation?⍭ Generate me a similar question◊

Sia