Suppose f satisfies the differential equation dy/dx=y+1 and f(0)=1. In other words, f′(x)=f(x)+1 and f(0)=1.
Use Euler's method with a step size of 0.5 to approximate f(1.5).
To use Euler's method, we will take small steps along the x-axis to approximate the y-values of the solution to the differential equation.
Given the differential equation dy/dx = y + 1 and the initial condition f(0) = 1, we can start by evaluating the derivative at x = 0:
f'(0) = f(0) + 1 = 1 + 1 = 2.
Now, we can use this information to approximate the value of f at x = 0.5 using a step size of 0.5:
f(0.5) ≈ f(0) + f'(0) * (0.5 - 0)
= 1 + 2 * 0.5
= 1 + 1
= 2.
We now have the approximate value of f at x = 0.5. Next, we can repeat this process to find the approximate value of f at x = 1:
f(1) ≈ f(0.5) + f'(0.5) * (1 - 0.5).
To find f'(0.5), we need to evaluate the differential equation at x = 0.5:
f'(0.5) = f(0.5) + 1.
We already found f(0.5) to be 2, so:
f'(0.5) = 2 + 1
= 3.
Substituting this back into the equation for f(1):
f(1) ≈ 2 + 3 * (1 - 0.5)
= 2 + 3 * 0.5
= 2 + 1.5
= 3.5.
Therefore, using Euler's method with a step size of 0.5, we can approximate f(1.5) to be 3.5.