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Let y=f(x) be the solution to the differential equation (dy)/(dx)=1+2y 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?

Let y=f(x)y=f(x) be the solution to the differential equation dydx=1+2y\frac{dy}{dx}=1+2y with the initial condition f(0)=1f(0)=1. What is the approximation for f(1)f(1) if Euler's method is used, starting at x=0x=0 with a step size of 0.50.5?

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Q. Let y=f(x)y=f(x) be the solution to the differential equation dydx=1+2y\frac{dy}{dx}=1+2y with the initial condition f(0)=1f(0)=1. What is the approximation for f(1)f(1) if Euler's method is used, starting at x=0x=0 with a step size of 0.50.5?
  1. Given Differential Equation: Given the differential equation (dydx)=1+2y(\frac{dy}{dx}) = 1 + 2y and the initial condition f(0)=1f(0) = 1. We need to find f(1)f(1) using Euler's method with a step size of 0.50.5.
  2. Calculate Number of Steps: Calculate the number of steps needed to reach x=1x = 1 from x=0x = 0 with a step size of 0.50.5. Number of steps = (1−0)/0.5=2(1 - 0) / 0.5 = 2.
  3. Start with Initial Values: Start with the initial values: x0=0x_0 = 0 and y0=1y_0 = 1.
  4. Calculate Slope at Initial Point: Calculate the slope at the initial point using the differential equation. f(x0,y0)=1+2y0=1+2(1)=3f(x_0, y_0) = 1 + 2y_0 = 1 + 2(1) = 3.
  5. Use Euler's Method for y1y_1: Use Euler's method to find y1y_1. y1=y0+step_size×f(x0,y0)y_1 = y_0 + \text{step\_size} \times f(x_0, y_0). y1=1+0.5×3=2.5y_1 = 1 + 0.5 \times 3 = 2.5.
  6. Update x to Next Value: Update x to the next value. x1=x0+step_size=0+0.5=0.5x_1 = x_0 + \text{step\_size} = 0 + 0.5 = 0.5.
  7. Calculate Slope at New Point: Calculate the slope at the new point (x1,y1)(x_1, y_1). f(x1,y1)=1+2y1=1+2(2.5)=6f(x_1, y_1) = 1 + 2y_1 = 1 + 2(2.5) = 6.
  8. Use Euler's Method for y2y_2: Use Euler's method to find y2y_2. y2=y1+step_size×f(x1,y1)y_2 = y_1 + \text{step\_size} \times f(x_1, y_1). y2=2.5+0.5×6=5.5y_2 = 2.5 + 0.5 \times 6 = 5.5.
  9. Update x to Next Value: Update x to the next value. x2=x1+step_size=0.5+0.5=1x_2 = x_1 + \text{step\_size} = 0.5 + 0.5 = 1.

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