Final answer:
To sketch the slope field and estimate the solution of a differential equation, we can use Euler's method with a given step size. The actual solution can then be compared with the estimated solution to determine if it is an over or under-estimate. In this case, the actual solution is y = 0.5 * e^(3x) which satisfies the differential equation y' = 3y and initial condition y(0) = 0.5.
Step-by-step explanation:
To sketch the slope field for the given differential equation, we need to find the slope at various points on the x-y plane. For the equation y' = 3y, the slope at any point (x, y) is 3y. We can choose several points on the x-y plane, calculate the slope at each point, and then plot arrows with those slopes.
Using Euler's method with a step size of h=0.2, we can estimate the solution at x=0.2, 0.4, ..., 1. Starting with the initial condition y(0) = 0.5, we can use the formula y(x+h) = y(x) + h * f(x, y(x)) to calculate the estimates.
Finally, we can plot the estimated solution on the slope field and compare it with the actual solution. Based on the given differential equation y' = 3y, the actual solution is given by y = 0.5 * e^(3x). We can check that this equation satisfies the given differential equation and initial condition.